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Derivation and practical application of exact time domain solutions for diffraction of acoustic waves… Dalton, David Raymond 1987

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D E R I V A T I O N A N D P R A C T I C A L A P P L I C A T I O N O F E X A C T T I M E D O M A I N S O L U T I O N S F O R D I F F R A C T I O N O F A C O U S T I C W A V E S B Y A H A L F P L A N E . B y D a v i d R a y m o n d D a l t o n B.Sc. M e m o r i a l Un i v e r s i t y of N e w f o u n d l a n d , 1985 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D e p a r t m e n t of Geophys i c s a n d A s t r o n o m y We accept th i s thesis as c o n f o r m i n g to the requ i red s t anda rd T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A December 1987 © D a v i d R a y m o n d D a l t o n , 1987 In presenting this thesis in partial fulfilment of the re-quirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is un-derstood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geophysics and Astronomy The University of British Columbia 1956 Main Mall Vancouver, Canada ABSTRACT. T h e h i s t o r y of d i f f rac t ion theory, exact f requency d o m a i n so lu t ions and se lected past t i m e d o m a i n so lut ions are br ie f l y rev iewed. E x a c t t ime d o m a i n so lu t ions fo r s ca t te r i ng of p l ane , c y l i n d r i c a l and sphe r i ca l acoust ic waves by a ha l f p l ane are de r i ved by inverse F o u r i e r t r a n s f o r m i n g the f requency d o m a i n in tegra l so lut ions . T h e so lut ions cons ist of two d i f f r a c t i on te rms , a ref lected t e r m and a d i rect t e r m . T h e d i f f r a c t i n g edge induces s tep f unc t i on d i s cont inu i t i e s in the d i rect and ref lected te rms at two shadow bounda r i e s . A t each bounda ry , the assoc iated d i f f r ac t i on t e r m reaches a m a x i m u m a m p l i t u d e of ha l f the geomet r i ca l opt ic s t e r m and has a s i g num f unc t i on d i s cont inu i t y , so t h a t the t o t a l field rema in s cont inuous . A phy s i ca l i n t e r p r e t a t i on is deve loped i n t e rms of Huygen ' s p r i n c i p l e near the d i f f r ac t i ng edge. So lu t i on s for p r a c t i c a l po i n t source conf igurat ions are eva lua ted by n u m e r i c a l l y c o n v o l v i n g the impu l se d i f f r ac t i on responses w i t h a wavelet. T h e n u m e r i c a l p rob lems presented by convo lu t i on w i t h a s ingu lar , t r u n c a t e d ope ra to r are so lved by a n a l y t i c a l l y de r i ved co r r e c t i on techn iques , w h i c h are f avou rab l y c o m p a r e d to those used by ear l ier au thor s . T h e d i f f r ac t i on s o l u t i on col lapses into a c o m p a c t d i s c re t i zed f o r m u l a t i o n . T h e ha l f p lane is shown to be a l i m i t i n g f o r m of wedge so lu t ions , w h i c h can thus be c o m p u t e d us ing s im i l a r a l go r i t hms . T w o zero offset sect ions are p r o d u c e d a n d c o m p a r e d to a p p r o x i m a t e K i r c h h o f f i i i n teg ra l so lut ions . T h e exact d i f f r ac t i on h y p e r b o l a is no t i ceab l y non - s ymmet r i c , w i t h h i gher amp l i t ude s on the ref lector side of the edge. Nea r the apex of the h y p e r b o l a t he K i r c h h o f f s o l u t i on is near ly equ iva lent to the exact d i f f r ac t i on t e r m s y m m e t r i c i n a m p l i t u d e about the ref lect ion shadow bounda r y bu t fai ls to descr ibe the other , low a m p l i t u d e , t e r m equ iva lent to ha l f the response of a l ine scatterer . T h e differences are more not i ceab le on the flanks of the h ype rbo l a , where the two terms are comparab le i n a m p l i t u d e , a n d at sha l low depths , due to an aper tu re effect. Increas ing e i ther the d e p t h of the edge or the angle of the se i smic l ine to the n o r m a l to the edge resu l t s i n a f l a t te r d i f f r ac t i on h y p e r b o l a showing l i t t l e a m p l i t u d e v a r i a t i o n w i t h moveout . A s the se i smic l ine becomes pa ra l l e l to the edge the d i f f r ac t i on curve becomes flat and is i nd i s t i ngu i shab le f r o m a re f lect ion event. A t great d e p t h d i f f r a c t i on events may be m i s t aken for ref lect ion events as we l l . E x a m p l e s of C D P and C S P gathers, when compa red to the C o m m o n Offset ( C O ) gathers , demons t r a te t h a t C O gathers are o p t i m a l for d i f f r ac t i on process ing. A l s o , s ince the d i f f r ac t i on moveou t and ref lect ion moveout curves differ w ide l y except for d e p t h po int s near the edge, n o r m a l moveout s t a c k i n g w i l l d i s t o r t the d i f f ract ions and d i f f r ac t i on s tack ing is essent ia l to r e t a i n d i f f r ac t i on i n f o r m a t i o n . S t r ip s of v a r y i n g w i d t h are m o d e l l e d by s upe rpo s i t i on of ha l f p lanes to demons t ra te r e so l u t i on effects a n d show t ha t the l i m i t of a s t r i p is a l ine scatterer. A d i p p i n g s t r i p a n d an offset ha l f p lane m o d e l are p r o d u c e d and added for l a ter c ompa r i s on w i t h wedge so lu t ions . i i i TABLE OF CONTENTS Abstract. ii List of Figures vii Acknowledgements. ix 1 Introduction. 1 1.1 History of Diffraction Theory 1 1.2 Seismic Diffraction Theory 7 1.3 Thesis Outline 9 1.4 Notations used throughout the thesis 10 2 Time domain response for various sources. 14 2.1 Generalized derivation procedure 14 2.2 Plane wave incidence 18 2.3 The line source problem 23 2.4 The Point Source Problem 29 2.5 Dipole Sources 37 iv 3 Comparison with other solutions. 39 3.1 P l a n e wave so l u t i on compar i sons 39 3.2 L i n e source s o l u t i on ve r i f i ca t i on 40 3.3 P o i n t source compar i sons 42 3.3.1 V e r i f i c a t i o n in the p lane wave l i m i t 42 3.3.2 C o m p a r i s o n w i t h W a i t ' s step f unc t i on so lu t i on 43 3.3.3 C o m p a r i s o n w i t h K i r c h h o f f theory 48 3.3.4 C o m p a r i s o n w i t h wedge so lut ions 49 4 Pract ical application. 53 4.1 O v e r v i e w 53 4.2 H a n d l i n g the s i n gu l a r i t y 54 4.3 T r u n c a t i o n " g h o s t " effect remova l 58 4.4 D i s c r e t i z ed f o r m of the convo l u t i on in tegra l 60 5 Examples. 63 5.1 S ing le t race examples 63 5.2 E x a m p l e sect ions w i t h K i r c h h o f f c ompa r i s on 69 5.3 O b l i q u e inc idence examp le s 76 5.4 D e p t h v a r i a t i o n 78 5.5 T r a c e gather v a r i a t i o n 80 5.6 S t r i p r e so l u t i on examp le s 82 5.7 F a u l t s i m u l a t i o n 84 v 6 C o n c l u s i o n s . 6.1 Point summary 6.2 Discussion 6.3 Areas for future research R e f e r e n c e s . L I S T O F F I G U R E S 1 H u y g e n s ' p r i n c i p l e app l i ed to edge d i f f r ac t i on 3 2 G e o m e t r i c a l t heo ry of d i f f rac t ion 6 3 Geomet r i e s for the three types of sources 17 4 G e o m e t r i c a l c on s t r u c t i o n for ve r i f y i n g the p lane wave d i rec t a n d re-flected a r r i v a l t imes 24 5 N u m e r i c a l check of the I F T of the Fresne l i n tegra l 41 6 S chemat i c rep re senta t i on of c onvo l u t i on of a wavelet w i t h the d i f f rac -t i o n ope ra to r 62 7 T h e wavelet a n d samp le d i f f rac t ion operator s used i n ca l cu l a t i on s . . . 65 8 C o m p a r i s o n of convo lu t ions pe r fo rmed w i t h and w i t h o u t the s i n gu l a r i t y c o r r e c t i on 66 9 C o m p a r i s o n of convo lu t ions cor rected a n d unco r rec ted for t r u n c a t i o n effects 68 10 K i r c h h o f f - e x a c t t o t a l field compar i sons for an edge at a d e p t h of 750 m. 70 11 D i f f r a c t ed field compar i sons for the m o d e l of F i g u r e 10 70 12 A s for F i g u r e 10, b u t for a ha l f p l ane at a d e p t h of 225 m 73 13 (Over lea f ) A s for F i g u r e 11, bu t for a ha l f p lane at a d e p t h of 225 m. 73 v i i 14 Schemat i c dep i c t i o n of a the response of a l ine scatterer 75 15 V a r i a t i o n s i n m o d e l ca l cu la t i on s as the angle of the seismic l ine w i t h the n o r m a l t o the edge is va r ied f r o m zero 77 16 V a r i a t i o n s in the m o d e l ca l cu la t ions w i t h rea l i s t ic depth and ve l oc i t y v a r i a t i on 79 17 C o m p a r i s o n of va r ious types of gathers and s tack ing p rocedures . . . . 81 18 Schemat i c dep i c t i o n of a the response of a s t r ip 82 19 S t r i p mode l s of v a r y i n g w id th s 83 20 Sect ions for a d i p p i n g s t r i p , offset ha l f p lanes and the s u m of these two mode l s 85 v i i i ACKNOWLEDGEMENTS. F i r s t and fo remost , I w i sh to t hank m y superv i so r , M a t t Y e d l i n , for his i n s p i r a t i ona l , enthus ia s t i c app roach to m a t h e m a t i c a l geophys ics , a n d for his pat ient gu idance and ass i s tance over the last e ighteen month s . M a n y t h a n k s are due to a l l member s of the U B C D e p a r t m e n t of Geophys i c s a n d A s t r o n o m y for m a k i n g m y stay here an enjoyable a n d en l i gh ten ing one, w i t h m u c h i n t e r a c t i on i n a cademic , soc ia l and a t h l e t i c arenas. Use fu l suggest ions, d iscuss ions a n d e d i t o r i a l c omment s were c o n t r i b u t e d by E d J u l l , B a r r y N a r o d , Bob E l l i s , A n d y C a l v e r t , Dave L u m l e y , and J i m W a i t , i n add i t i on to my superv i sor . A mod i f i ed ve r s i on of Pe te r K r y s z k i e w i c z ' s M A T H E S I S sty le for-m a t for the IATgX t ypese t t i ng p r o g r a m , coup led w i t h R o b e r t P i c a r d ' s p rocedu re for i n se r t i ng g raph ic s d i rec t l y i n to the t ex t , was used to p roduce th i s thesis on the U B C c o m p u t i n g centre ' s Q M S laser p r i n t e r . V a l u a b l e ass istance was p rov i ded by U B C l i b r a r y and U B C c o m p u t i n g cent re staff. M y stay at U B C was s u p p o r t e d by a N a t u r a l Sciences and Eng i nee r i n g Research C o u n c i l of C a n a d a ( N S E R C ) 1967 Scho la r sh ip , a n d , for the past two month s , by N S E R C s t ra teg ic grant 5-82021. Fund s for p h o t o c o p y i n g and for c o m p u t i n g on the A m d a h l were p rov i ded by N S E R C o p e r a t i n g g rant 5-80642. i x 1 . I N T R O D U C T I O N . 1.1 H i s t o r y o f D i f f r a c t i o n T h e o r y . D i f f r a c t i o n is the process by w h i c h wave p r o p a g a t i o n deviates f r o m the p red i c t i on s of r ay theory , or geomet r i ca l opt i c s . D i f f r ac t i on s m a y be ana lyzed qua l i t a t i ve l y , by the genera l i zed wave theory of acoust ics or e lec t romagnet i c s , or d i ssected quan t i t a t i v e l y by so lu t i ons of the wave equa t i on , the H e l m h o l t z equa t i on , or M a x w e l l ' s equat ions, sub ject to specif ic b o u n d a r y and/or i n i t i a l cond i t i ons . G e o m e t r i c opt i c s ( G O ) is an a p p r o x i m a t e h i gh f requency so lu t i on of the wave equa t i on , and d i f f r ac t i on demon -s t rates the fa i l u re of tha t a p p r o x i m a t i o n w h e n the wave leng th of a p r o p a g a t i n g wave is t he same order as cha rac te r i s t i c l ength scales of the s u p p o r t i n g m e d i u m . Ob se r va t i on of d i f f rac t i ve phenomena in n a t u r e and the a n a l y t i c a l s o l u t i on of d i f f r a c t i o n p rob lems have a long and i l l u s t r i ous h istory. T h e earl iest accurate de-s c r i p t i o n s t i l l ex tant is one by G r i m a l d i (1660), a l t hough there was some reference to d i f f r a c t i o n i n the works of L e o n a r d o d a V i n c i (1452-1519). Hooke (1665) also made a no te abou t d i f f r ac t i on . Isaac N e w t o n (1672) observed d i f f r ac t i on b u t ma in t a i ned t h a t it d i d not c on t r ad i c t the p a r t i c u l a t e theory of l ight. T h i s v i e w p o i n t is reason-ab le , s ince d i f f ract ions c a n be p a r t i a l l y e xp l a i ned i n te rms of p a r t i c l e inc idence u p o n a d i s cont inu i t y . Fo r e xamp le , a s l ight v a r i a t i o n i n the angle of i nc idence of a p o o l ba l l 1 on a b u m p e r p roduces a w ide va r i a t i on in scat te r ing angle. I ron ica l l y , Huygen s (1690), wh i l e he advanced a wave theo ry of l i ght , d i d not accept d i f f r a c t i on as p a r t of i t . These v i ewpo in t s h i nde red progress i n d i f f r a c t i on theory for over a century . T h e n , Y o u n g (1802) advanced a theory of d i f f rac t ions emana t i n g f r o m the edge of a bar r ie r , wh i ch acts as a secondary source. F resne l (1816), f o l l ow ing Huygen s ideas of secondary wavelets, p o s t u l a t ed t ha t the d i f f r ac t i on resu l ted f r o m the ba r r i e r o b s t r u c t i n g some of the secondary wavelets, w i t h the r e m a i n i n g in te r fe r ing to p r oduce the d i f f r a c t i on p a t t e r n . H i s was the first m a t h e m a t i c a l ana lys i s of d i f f r ac t i on b u t was not r igorous. It led to the no t i on of F resne l zones. Fresne l ' s ideas p reva i led over Young ' s for nea r l y 150 years. A few decades la ter the wave theory of d i f f r ac t i on was un i ve r sa l l y accepted. In-deed, it was necessary to show how long d i s tance r ad i o c o m m u n i c a t i o n was poss ible. N u m e r o u s app l i ed m a t h e m a t i c i a n s then began to ca l cu la te d i f f r ac t i on pa t te rn s p r o -d u c e d ' b y s imp le geomet r i ca l objects. K i r c h h o f f (1882) p u t Fresne l ' s ana lys i s on a s ound m a t h e m a t i c a l basis, express ing Huygen ' s p r i n c i p l e in te rms of an in tegra l t he -o rem. However , his results were only a p p r o x i m a t e l y cor rect , due t o the a p p r o x i m a t e b o u n d a r y cond i t i on s used, w h i c h set the t o t a l f ie ld and its de r i va t i ve equa l to the i n -c ident f ie ld and its der i vat i ve in the i l l u m i n a t e d zone a n d to zero in the shadow zone. A n excel lent t r e a tmen t of the m a t h e m a t i c a l t heo ry of Huygen ' s p r i n c i p l e , showing t h a t the d i f f rac ted f ie ld becomes equiva lent to the geomet r i ca l opt i c s f ie ld as the edge is app roached , a n d c r i t i c i s i n g K i r c h h o f f ' s a s sumpt ions , is g i ven i n B a k e r and C o p s o n (1939). U s i n g expans ions most eas i ly eva luated w h e n the p r o d u c t of l ength scale (a) a n d wavenumber (k) is s m a l l , K i r c h h o f f o b t a i n e d an a p p r o x i m a t e s o l u t i on to the 2 (a) PLANE WAVEFRONT (b) WAVEFRONT AT AN EDGE Figure 1: S chemat i c rep re senta t i on of H u y g e n s ' p r i n c i p l e as app l i ed to edge d i f f rac -t i o n , (a) A cont inuous re f lect ion as a s u m of wavefronts , (b) A t r u n c a t e d ref lector w i t h some of the wavelets f o r m i n g a d i f f rac ted wave. ( R e d r a w n f r o m W r e n (1987).) 3 d i f f r a c t i o n by a ho le in a screen, Ray l e i g h (1897) so lved the case of d i f f r ac t i on by a pe r f e c t l y c o n d u c t i n g c y l i nde r , and numerous others ob ta i ned s im i l a r results for the sphere, s t r i p , c i r c u l a r d isk, semi - in f i n i te cone a n d o ther p rob lems . However , none of these are easi ly eva luated w h e n the p r o d u c t ka is large. Som-m e r f e l d (1896) devised a new m e t h o d of so l v ing the p r o b l e m of s ca t te r i ng of a p lane wave by a ha l f p lane. H i s s o l u t i on i nvo l ved c o m p l e x i n t e g r a t i o n over a two-sheeted R i e m a n n space w h i c h y i e l ded the so lu t i on in te rms of Fresnel integrals . Un f o r t una te l y , t h i s m a y have h i nde red progress, because many peop le t r i ed to app l y these techniques to o the r p rob l ems over the next 50 years, to no ava i l . A t t e n t i o n was t hen focussed on h igh f requency a s ympto t i c so lu t ions by Peker i s , Fock a n d others , a n d on Lunenbe r g ' s lectures i n the 1940's on the m a t h e m a t i c a l t heo r y of geomet r i ca l opt i c s . T h e forego ing results set the stage for K e l l e r (1962) and his col leagues to develop the geomet r i ca l t heo ry of d i f f r a c t i on ( G T D ) , w h i c h assumes tha t d i f f r ac t i on is a loca l effect and t ha t away f r o m the edge the d i f f rac ted field behaves l ike d i f f rac ted rays. It e x tended F e r m a t ' s (1657) p r i n c i p l e t o define an edge d i f f rac ted ray as the one curve, a m o n g those con s t r a i ned to j o i n source and receiver v i a a po in t on the edge, t h a t has s t a t i o n a r y o p t i c a l l eng th . 4 For a ray n o r m a l l y i nc ident to a s t ra i ght edge t ha t edge acts as a l ine source w h i c h p roduces a c y l i n d r i c a l wave w i t h a m p l i t u d e mod i f i ed by an angle dependent d i f f r a c t i on coeff ic ient. If Ax is the inc ident f ie ld a m p l i t u d e , AD is the d i f f rac ted f ie ld a m p l i t u d e a n d D(4>0,<j>) is the d i f f r ac t i on coeff ic ient, t hen , re fe r r ing to F i g u r e 2, AD = AiD{d>Q,<f>)^. (1) A g a i n re fe r r i ng t o F i g u r e 2, the d i f f r a c t i on a m p l i t u d e s are largest i n the pa rabo l i c regions R (wh i ch has the re f lect ion b o u n d a r y zone as its axis) a n d S (wh ich has the shadow zone b o u n d a r y as its ax i s ) , a n d i t is i n prec i se ly these regions tha t the G T D s o l u t i on breaks down . T h u s the G T D is not app l i cab le to seismic d i f f ract ions. Ins tead, near the bounda r i e s of the geomet r i ca l opt i c s f ie ld or near the edge more r igorous so lu t ions mus t be used. A h igh- f requency u n i f o r m a s y m p t o t i c theory ( U A T ) of d i f f r ac t i on was deve loped by Lew i s a n d B o e r s m a (1969) to hand le the s ingu lar i t ies i n G T D . N u m e r o u s o ther au tho r s have p roposed va r ia t ions or extens ions t o this the -ory. E x a c t so lu t ions have not been cons idered due to the i r l i m i t e d scope and great c o m p l e x i t y for a l l bu t a few s imp le cases. 5 Figure 2: Schematic representation of the geometrical theory of diffraction, showing the parabolic regions in which GTD fails. The reflection and direct shadow boundaries are R and S. The reflected illuminated zone is 1, the direct is 1 + 2 and 3 is the shadow zone, in which only diffractions are observed, (after Jull, 1981). 6 1 . 2 Seismic Diffraction Theory. W h y is i t i m p o r t a n t to re -examine c lass ica l p rob lems such as t ha t of d i f f r ac t i on by a ha l f p lane? In seismology, a t i m e d o m a i n acoust ic s o l u t i on is preferable t o the u sua l f requency d o m a i n so lu t i on . A l s o , exact so lut ions serve b o t h as a check aga inst fa r f ie ld so lu t i ons and a p p r o x i m a t e so lu t ions such as K i r c h h o f f theory, w h i c h does no t sat i s fy the correct b o u n d a r y cond i t i on s , and serve as a basis for more comp lex s tud ies . W i t h t r ue a m p l i t u d e and phase recovery, d i f f r ac t i on i n f o r m a t i o n m a y be used to con s t r a i n mode l s a n d poss ib ly even infer some of the geology at r ight angles to the seismic survey. E x t r e m e l y h i gh re so lu t i on is requ i red t oday to detect s m a l l s t r u c t u r a l t raps a n d s t r a t i g r aph i c t raps . D i f f r a c t i on s can result f r o m features t o o s m a l l to p roduce ref lect ions a n d w i t h app rop r i a t e process ing c ou l d be used t o bo l s ter or d i spe l hopes o therwi se based on ly on subt le wave f o rm phase changes. T h e h y p e r b o l i c " i n ve r t ed smi le s " or "bow t ie s " of d i f f r ac t i on pa t te rn s , w h i c h have been ev ident on seismic re f lect ion sect ions s ince the techn ique began, are p r o d u c e d by a b r u p t changes in m a t e r i a l p roper t i e s and thus may p rov ide i n f o r m a t i o n on the t ype of d i s con t i nu i t y and its l o ca t i on . T h e y may s igna l a ho r i z on t a l change i n impedance or a ref lector t e r m i n a t i o n a n d he lp ease the i den t i f i c a t i on of fau l t s , channe l s , u n c o n -fo rm i t i e s , reefs and "b r i gh t s po t s " . W i t h recent advances i n acqu i s i t i on and process ing techn iques, reco rded seis-m i c d a t a approaches t r ue a m p l i t u d e a n d phase recovery. T h u s the subt le wave f o rm changes t h a t occu r a long se ismic d i f f r ac t i on t r a ve l - t ime shou ld be re -ana l yzed , a n d any theor ies app l i ed to d i f f r a c t i on process ing e x a m i n e d for poss ib le errors or def i c ien-cies a n d rev i sed if necessary. 7 Instead of t r ea t i n g d i f f rac t ions as noise a n d a t t enua t i n g t h e m , they cou l d be ana -l y zed separate ly to p rov i de a d d i t i o n a l cons t ra in t s on re f lect ion se ismic mode l s . C D P s t a c k i n g fo l l owed by m i g r a t i o n m i n im i ze s d i f f rac t i ve i n f o rma t i on wh i le enhanc i ng re-flection i n f o r m a t i o n . Pe rhap s another f o r m of process ing cou ld o p t i m a l l y enhance re t r i eva l of i n f o rma t i on f r o m d i f f ract ions . One poss ib le d i f f r ac t i on process ing m e t h o d , i n v o l v i n g a p p l i c a t i o n of phase co r re l a t i on and s t a t i s t i ca l tests, is o u t l i n ed by L a n d a et a l . (1987). A n o t h e r m e t h o d , i n vo l v i n g a d i f f r ac t i on or " c o m m o n fau l t p o i n t " ( C F P ) s t ack i ng , is descr ibed by P h a d k e and K a n a s e w i c h (1987). E x a c t d i f f r a c t i on so lut ions s hou l d resu l t i n improvement s in these fau l t l o ca t i on techniques and i n m i g r a t i o n rout ines . D u e to the seismic s u r vey i ng l im i t s on source and receiver po s i t i on s su rvey ing techn ique and due to the geolog ica l l im i t s on the types of d i f f rac to r geometr ies con -s ide red , the h i s to ry of se i smic d i f f r a c t i on theo ry is not near l y as c o m p l i c a t e d as t h a t of e l ec t romagnet i c d i f f r a c t i on theory, w h i c h was br ie f ly rev iewed i n Sect ion 1.1. A more comprehens i ve rev iew was w r i t t e n by K e l l e r (1985). K r e y (1952) and K u n z (1960) showed t ha t d i f f r ac t i on of se i smic waves was ev ident i n prof i les across f au l t s , sudden facies changes or ref lector t e r m i n a t i o n s , a n d t h a t , a l t hough th i s wou l d a i d i n the i den -t i f i c a t i o n of some features, the t rue b o u n d a r y of the d i s con t i nu i t y a n d perhaps o ther features as wel l w o u l d be masked . De H o o p (1958) pub l i s hed a comprehes i ve thesis on e l a s t o d y n a m i c d i f f r ac t i on theo ry for d i f f r ac t i on of S H a n d P p lane waves by a ha l f p lane. Wo r k on wave e q u a t i o n m i g r a t i o n techn iques for r emov i n g d i f f r a c ted energy was begun in the 1970's by C l a e r b o u t (1970, 1976), S to l t (1978) and B e r k h o u t a n d P a l t h e (1976), among others . Recent d i f f r ac t i on m o d e l l i n g has been p e r f o r m e d by use of K i r c h h o f f d i f f r ac t i on 8 theory , n o t a b l y by T ro rey (1970, 1977), B e r r y h i l l (1977) a n d H i l t e r m a n n (1970, 1975). These papers descr ibe an a m p l i t u d e - s y m m e t r i c d i f f r a c t i on h y b e r b o l a w i t h its apex at the ref lector t e r m i n a t i o n , a po l a r i t y reversa l when the source-receiver m i d p o i n t crosses the t e r m i n a t i o n and amp l i t ude s con t ro l l ed by the l o ca t i on of the m i d p o i n t w i t h re-spect to the t e r m i n a t i o n . A n a l o g d i f f r ac t i on mode l l i n g , f i rst pe r f o rmed by A n g o n a (1960), was used by H i l t e r m a n n (1970) to test K i r c h h o f f so lut ions . E x p e r i m e n t a l re-su l t s o b t a i n e d by H i l t e r m a n n for s y m m e t r i c a l d i f f r ac to r con f i gu ra t i on s (eg. domes, sync l ines ) agreed we l l w i t h the K i r c h h o f f theory results . However , K i r c h h o f f so lu -t i ons for n o n - s y m m e t r i c a l d i f f rac to r conf i gurat ions (eg. fau l t s ) e x h i b i t e d not iceab le differences f r o m the e x p e r i m e n t a l results. 1.3 Thesis Outline. T h i s thesis rev iews var ious t ime a n d f requency d o m a i n so lu t ions for acoust i c d i f f rac-t i o n by a ha l f -p lane , l inks t h e m by a genera l i zed Fou r i e r t r a n s f o r m app roach , and presents t h e m in a genera l i zed f o rma t su i tab le for p r a c t i c a l a p p l i c a t i o n and compa r -i son to f ie ld da t a , a p p r o x i m a t e far - f ie ld so lut ions , K i r c h h o f f so lu t ions a n d prev ious exact t ime d o m a i n d i f f r a c t i on so lut ions by de H o o p (1958), Felsen a n d M a r c u v i t z (1973) , W a i t (1957) and B i o t and To l s toy (1957). T h i s f o r m a t , w h i c h lends itself t o p h y s i c a l i n t e r p r e t a t i o n , and the p r a c t i c a l a p p l i c a t i o n techn iques, w h i c h c an easi ly be e x tended to hand le genera l wedge so lut ions , are i n tended to make exac t d i f f rac t ion so lu t i ons more access ible and more read i l y unde r s t andab l e to se ismologists . In C h a p t e r 2 we w i l l rev iew k n o w n f requency d o m a i n so lu t ions for d i f f r ac t i on of p l ane , c y l i n d r i c a l and sphe r i ca l waves by a ha l f p lane a n d inverse Fou r i e r t r a n s f o r m 9 t h e m to o b t a i n co r re spond ing t ime d o m a i n pu l sed source so lut ions. These so lut ions are presented in a f o r m amenab le to phy s i ca l i n t e rp re ta t i on i n terms of step and s i g n u m func t i on s and i n terms of Huygen ' s p r i n c i p l e near the d i f f r a c t i ng edge. A l s o , known t i m e d o m a i n so lut ions, der ived by other methods , for d i f f r ac t i on by a ha l f p lane or a wedge w i l l c ompa red to our so lut ions in C h a p t e r 3. C o n v o l u t i o n of the p lane wave and po in t source impu l se so lut ions w i t h source wavelets w i l l y i e l d t i m e d o m a i n d i f f r ac t i on so lut ions for rea l i s t ic seismic sources. E x -tended sources may be s imu la ted by a spa t i a l i n teg ra t i on of po i n t source so lut ions. N u m e r i c a l p r ob l ems re la ted to the convo l u t i on w i l l be t ack l ed in C h a p t e r 4 a n d c o m -pa red t o those used by B e r r y h i l l (1977). In C h a p t e r 5 two zero offset po i n t source examples w i l l be p r oduced and c o m -pa red t o those p r o d u c e d u s ing a p p r o x i m a t e theories for s im i l a r conf igurat ions . A l s o , we w i l l e x am ine var ious types of t race gathers. A deep example , d i p p i n g examp le , s t r ip s o f two w i d t h s , offset ha l f p lanes, a step and a wedge w i l l also be mode l l ed by s u p e r p o s i t i o n of ha l f planes w i t h app rop r i a te as sumpt ions . C o m p a r i s o n w i l l be made w i t h s i m i l a r examples p r o d u c e d by ear l ier author s u s ing a p p r o x i m a t e so lut ions and w i t h e x p e r i m e n t a l resu lts . A d i scuss ion of the s ign i f icance of our resu l t s in te rms of se i smic i n t e r p r e t a t i o n w i l l be made. 1.4 Nota t ions used throughout the thesis. In order to m i n i m i z e redundanc ie s , we w i l l now define nota t ions and a s sumpt ions for use in subsequent der i vat ions . Def ine the s i gnum f u n c t i o n , step f u n c t i o n and s ine, 10 cos ine a n d exponen t i a l Fresnel integra l s as sgn(x) H(x) -1.0, x < 0 0.0, x = 0 I 1.0, x > 0 ( 0.0, x < 0 1 + sgn(x) 0.5, x = 0 , I 1.0, x > 0 1 c o s V C(x) = \ — [ c o s t>2 = ^ — f \l n Jo V ^ T T io , . [2 fx . , , 1 p 2 sin u S i x ) = \/ — / s i n i r dv — ,— / —— dv V 7T io y/2n Jo Jv and F±{x) = J-{C ±iS)= e±lu~ du F±(x) + F±{-x) = yfit V 2 ^1 (2) (3) (4) (5) (6) Represent the i/^ -order H a n k e l f unc t i on s of the f irst and second k i n d by and H(2\ Deno te the acoust ic ve l oc i t y by c, the wavenumber by k = OJ/C a n d the acoust ic ve l o c i t y p o t e n t i a l by V. Def ine an acous t i ca l l y r i g i d (or ha rd ) ha l f p lane as one w h i c h has a zero pressure de r i va t i ve on i ts surface, a n d an acous t i ca l l y soft (or weak) ha l f p lane as one t ha t has zero pressure on its surface. D i f f r a c t i o n of acoust ic waves by a r i g i d b o d y is equ iva lent to d i f f r ac t i on of a t ransverse magne t i c ( T M ) po l a r i zed e l ec t romagnet i c ( E M ) f ie ld by a per fect l y c o n d u c t i n g ha l f p lane, a n d d i f f r ac t i on of acou s t i ca l waves by a per fect l y soft ha l f p lane is equ iva lent to t ransverse e lectr ic ( T E ) p o l a r i z e d E M d i f f r ac t i on by a ha l f p lane. S ince a r b i t r a r i l y po l a r i z ed two -d imens i ona l ape r tu re E M fields m a y be decomposed in to T M a n d T E fields, the results der i ved here in for acoust i c d i f f r ac t i on are equa l l y app l i c ab l e t o e lec t romagnet i c d i f f r ac t i on . 11 We sha l l assume that V(—u>) = V*(u>), where * impl ies comp lex con juga t i on , so t h a t the t i m e d o m a i n so lu t i on w i l l be rea l -va lued. A l t e r na t i v e l y , i f V__(u)) is the d i f f r a c t i on s o l u t i on for a t i m e dependence e _ U J ' a n d V+ (LO) is the s o l u t i on for a t i m e dependence e ' u t , t h e n V+(u>) = V*(LO). R e p l a c i n g i by — i i n e i ther f o r m of the so lu t ion resu l t s in the o the r f o rm. T h i s ensures tha t the f o l l ow i ng f o r m of the inverse Four ie r t r a n s f o r m ( I F T ) holds. 1 r°° ~ 1 r°° ~ 1 r°° ~ V(t) = — V{to)e-,u>t du = - R e / V_(w )c- , h r t dw = -Re / V+(u)elutdu. (7) Z 7 T J-oo 7 T JO TT Jo A s an as ide, we w i l l now check th i s for the p lane wave case. Fo r a t i m e depen-dence e~lwt, /•co V_(w) a e - " / 4 e ™ * * i / e t u du, w > 0 ( B o w m a n et a l . , 1969). (8) J — s/2watf>2 T h e pa ramete r s <f>i, fa and a, w h i c h w i l l be def ined i n te rms of the p r o b l e m geometry in Sec t i on 2.2, are not i m p o r t a n t in th i s d i scuss ion. F o r a t i m e dependence elu>t, the f o r m be low appl ies. /"CO , 7 + ( u ) a e " V ' ^ 1 _ e~iu" du, w > 0 ( J u l l , 1981). (9) •/ -\/2ioait>2 Fo r w < 0, a t ime dependence e~twt = e ' ^ l ' , a n d hence = VV ( - u ; ) oc e"/*eiua*i / e ~ , u " = V_* ( -w) (10) •/ — -\/—2 i d at (£2 T h u s the I F T in tegra l becomes 1 r°° . ~ l r r°° ~ r° — / c - t h " V ( w ) d u ; = — / e-xwiV_(uj) du + / e - , t " V r + ( - w ) d( 27T ./-co 27T L/o ./-co = ^ / f [e - * - "V_ (w) + e i w l 7 + ( « ) ] du; = - R e / e ~ , u ; t F _ ( w ) d u ; = - R e / elwtV+{u) du (11) 7T Jo TT Jo 12 In the last s tep the r e l a t i on = V2(w) has been used. T h i s ho lds for the l ine and po i n t source so lu t ions as we l l a n d w i l l be a s sumed in fu tu re der ivat ions . In the rest of the thes is , assume a h a r m o n i c t i m e dependence e~thlt, i.e. beg in w i t h the f requency d o m a i n ve l oc i t y po t en t i a l s o l u t i on V[u>) = V-{OJ) a n d use the f o l l ow ing f o r m of the F o u r i e r t r a n s f o r m pa i r . A l s o note t h a t the D i r a c d e l t a f unc t i on forms a spec ia l case of the above, as fo l lows. oo 1 f°° (12) — oo (13) T h i s r e l a t i on w i l l p rove usefu l i n many of the later der i vat ions . 13 2 . TIME DOMAIN RESPONSE FOR VARIOUS SOURCES. 2.1 Generalized derivation procedure. From inspection of various solutions, the details of which are left for later analysis, the generalized total-field time-domain for a plane wave, line or or point source can be represented as VT(u>) = Vi(w) ± V2H (14) where ± <-> ^ t d acoustic boundary conditions for the half plane, Vi(w) is the total field component due to the source and V2(OJ)IS the total field due to the image of the source across the half plane. Consider any of the geometries depicted in Figure 3, and make the definitions cos((p - <f>0), A2 = cos(0 + 4>Q) <f> - <t>o = H cos hi Si = sgn l cos H cos s 2 = s g n I c o s 2 4> - 4>o 2 <f> + <t>o 2 4> + <t>o = . H ( 7 r - | ^ - ^ 0 | ) , = sgn(?r - \d> - 0o|), = H(TX - \<f> + 4>0\) and | = sgn(?r - \<j> + (b0\). (15) (16) (17) (18) (19) 14 T h e re f lec t ion shadow bounda r y occurs at the angle <f> — TT — <j>o a n d the d i rec t shadow bounda r i e s occurs at an angle <f> = 7r + <p0, so tha t the re f lect ion t e r m w i l l be mod i f i ed by h2, the d i rec t by hi and two assoc iated d i f f r ac t i on te rms by s2 a n d T h e general s o l u t i o n (14) may thus be s u b d i v i d e d into geomet r i ca l opt i c s a n d d i f f r ac ted terms, u s i ng t he fo l l ow ing general f o r m of the componen t te rms , deduced by i n spec t i n g the d i f f r a c t i on so lut ions for var ious sources. ~ roo roo roo Vi(ui) — / fi(u2,UJ) du = 2hi / / i ( u 2 , (JJ ) du — Si / fi{u2,u) du Jai\mi\ JO J\mi\ = hiVGl(u)-SiVDl(u) (20) „ roo roo r oo V2(UJ) = / f2(u2,oj)du = 2h2 f2(u2,uj) du - s2 f2(u2,oj)du J Sn\m.2\ JO ^ |m.21 = h2VGi{tj)-s2VD2{uj). (21) In the above, m j and m2, wh i ch are e xp l i c i t l y def ined in later sect ions, are p r o p o r t i o n a l to y/oJ a n d also con ta i n an angle dependence, for a l l types of sources cons idered. T h e pa ramete r s m\ and m2 b o t h app roach zero at the edge a n d mi —» 0 at the d i rec t shadow b o u n d a r y a n d m2 —•»• 0 at the ref lected shadow bounda r y . T h u s for any angle of app roach , the t o t a l field remains cont inuous at the edge, and also at the shadow bounda r i e s . T h e genera l i zed t i m e d o m a i n response for a 6(t) source t i m e dependence is then de r i ved by inverse Fou r i e r t r an s f o rm ing (14), (20) a n d (21) to o b t a i n VT(t) = Vi(t)±V2(t), (22) V i ( * ) = M ' U O - S i W O , V2{t) = h2VGa{t) - s2VD7(t) (23) 15 VGl(t) = 2\\mVDl(t), VG,(t) = 2 l i m VD2{t), (24) m i — > 0 m s — » 0 where VGi is the d i rec t field, VDi is the assoc iated d i f f rac t ion t e r m w i t h a m a x i -m u m a m p l i t u d e at the d i rect shadow bounda ry , VG2 is the ref lected field and VDn is the as soc ia ted d i f f r a c t i on t e r m w i t h a m a x i m u m a m p l i t u d e at the re f lec t ion shadow bounda r y . Sect ions 2.2, 2.3 a n d 2.4 w i l l each use these re lat ions , desc r ibed above to avo id unnecessary r epe t i t i o n of steps, i n the f o l l ow ing manner : (a) T h e t e r m VDl(w) is deduced f r o m the f u l l s o lu t i on Vt(<JJ). (b) T h e t r a n s f o r m VDi(t) is eva luated. T h i s is the h a r d pa r t . (c) L i m i t s equ iva lent to those i n (24) are eva luated to o b t a i n VGl(t). (d) VGl(t) is added to VDl(t) to y i e l d V^t). (e) T h e second t e r m V2(i) is done ana logous ly to y i e l d VT(t) In s t anda rd se ismic process ing, the on ly t e rms cons idered are VQ2 and the approx -i m a t e equ iva lent of Vry2. However, the t e r m VD1 does make a s m a l l c o n t r i b u t i o n to the d i f f r a c t i on a m p l i t u d e s observed on se i smic sect ions and shou ld not be neglected. Re f e r r i n g t o the geometr ies i n F i g u r e 3 and the def in i t ions of the step and s i gnum func t i on s in Sect ion 1.4, a s imple phy s i ca l i n t e r p r e t a t i on of E q u a t i o n s (22) —(24) is poss ib le. R e s t r i c t i n g the d i scuss ion t o V2(t), the ref lected t e r m VQ2 has a d i s con t i nu -i ty across the re f lect ion shadow zone b o u n d a r y w h i c h is ma t ched by a d i s con t i nu i t y i n the d i f f r ac ted t e r m VD2, so t h a t the t o t a l f ie ld rema ins cont inuous . T h e ref lected 16 Figure 3: Geometry for (a) plane wave, (b) line source and (c) point source incidence on a half plane, (after Bowman et al. (1969) ) 17 field d rops f r o m a po s i t i ve va lue t o zero and the d i f f r ac ted field undergoes a po l a r i t y reversa l across the bounda r y . Fo r i l l u s t r a t i v e purposes assume the ref lected field has an a m p l i t u d e of 2 j u s t to the r i ght of the bounda r y . T h e n r ight on the b o u n d a r y it has an a m p l i t u d e of 1 and to the left it has one of 0. T h e assoc iated d i f f rac ted field Vjrj, has an a m p l i t u d e of —1 j u s t to the r i gh t of the bounda r y , 0 r i gh t on the b o u n d a r y and +1 j u s t to the left of the bounda r y . T h e t o t a l field rema in s cont inuous w i t h an a m p l i t u d e of 1. T h i s s imp l i s t i c i n t e r p r e t a t i o n w i l l be ex tended la ter w i t h concrete examples. A s im i l a r s i t ua t i on occur s for the d i r e c t /d i f f r a c ted s u m V ^ i ) near the d i rect shadow zone boundary . T h i s differs f r o m i n d u s t r y conven t i on , w h i c h assigns a va lue of 2 to the ref lected f ie ld a n d -1 to the d i f f r ac ted field r i ght on the bounda r y . T h i s cor responds to a s l i ght ly d i f ferent de f i n i t i on of the step a n d s i gnum func t i on s . 2.2 Plane wave incidence. C o n s i d e r a p lane wave i nc ident at an angle 4>o on a ha l f p lane. Re f e r r i n g t o F i g u r e 3(a), a n d l e t t i n g a — R/c, th i s p l ane wave acoust ic ve l o c i t y p o t e n t i a l is Vi(oj) = c - « * * « » ( * - * < . ) = e - . - " a c o . ( * - * „ ) ^ y.^ = 6(t + acos|</> - <&,]) (25) Fo r a receiver l oca ted at (R,<f>) w i t h respect to the edge, dep ic ted i n F i g u r e 3a, the t o t a l field f requency d o m a i n d i f f r ac t i on so lu t i on for soft (—) or r i g i d (+) b o u n d a r y 18 cond i t i on s is ( Sommer fe l d 1896) Vt(OJ) r/4 -\/2um COS '<f>- (j>0 ±e -1UO COS ip —0(| -y/2wct cos (26) S o m m e r f e l d (1896) and Ca r s l aw (1899) de r i ved the above equa t i on us ing the m e t h o d of m a n y - v a l u e d wave funct ions . C l e m m o w (1951), C o p s o n (1950) etc re -der ived the s o l u t i on u s ing a more concise m e t h o d i n vo l v i n g in tegra l equat i on a n d c o m p l e x con tou r i n t eg ra t i on techn iques M a c d o n a l d (1902) de r i ved an a l te rnate s o l u t i on in the f o r m of a n in f i n i te series. N o w pe r f o rm the inverse Four ie r t r an s f o rm of (26), u s i ng the no ta t i on s a n d procedures ou t l i ned in Sect ions 1.4 and 2.1, w i t h the added def in i t ions = V2 ua cos I — - — I and m 2 = V2u;acos I (27) to o b t a i n the fo l l ow ing express ions. VG(t) = — ^ R e c - X ' + " i i ) + * / 4 } ( 2 r e l u 2 d u } doj 7Ty/n Jo { Jo ) 1 roo = - R e / c-M«+°^i) du = 6{t + aAx). 7T Jo (28) VDl (t) = — ^ R e f°° c--{«(*+^i)+'/4} f°° e'"2 du dw TVy/n Jo J\fwa(\+Ai) = 1 R e f°° e - ^ + ^ ^ I ^ U u d u . •Ky/n Jo Jy/u)Ct{l+Al) (29) 19 L e t 7 = a(l - j - v i i ) and r = t -\- a A i , and reverse the order of i n teg ra t i on to ob ta i n TTy/TT JO JO OO f U ~ jt COS 7T , W7 H U 4 dw du = — f n^/nr Jo sm 2" — s in u .4 (30) S ince the second t e r m above reduces to 7= / s in I u 2 ) du = ; = ( / c o s f u 2 ) du — j s i n f u 2 ) du) = 0 , (31) a n d the f ac to r r t + aAx — oc — aAi t — a 1~ a(l + Ai) ~~ ft(l + Ai) (32) ^ (0 7T\/27rr i o [ cos (< - a ) u 2 a ( l + A x ) + s m (t - a)u2 a(l + A i ) du (33) We let t; = u\JJ^+Aj> w h i c h impl ies t h a t = u 2 s gn ( i — a ) , to o b t a i n V D l (0 = V ' , z / {cos(t; 2) + sgn(« - a) s i n ( ^ 2 ) } du [1 + sgn(* - a ) j = ^ A H(t - a). 27iT\/t — a (34) 20 A s a check, examine l i m VDi(t), where m i = \ ceil + y l O . T h i s l im i t is zero except for t — a , w h e n it is in f in i te . E x a m i n e its behav iou r as an i n teg rand as fol lows. lx = / 2 hm VDl{t) dt = l i m / — ] ^ . oo 1 Ul. (35) du 2 I - i / u — ^ _ = — h m < t a n A / u + m{\\/u 7r mj—o ^ y m i 2 m i /" * ,. i - i = l i m — / ^ — F = = — h m < t a n m,-tO 7f ,/rj T h u s the l i m i t of the d i f f r a c t i on t e r m is a d e l t a f u n c t i o n . N o t e t h a t l e t t i n g a —» 0 or Ai —> — 1 is equ iva lent to l e t t i n g m x —• 0. T h u s as the shadow b o u n d a r y is app roached , VDi —> 6(t — a)/2, in agreement w i t h E q u a t i o n s (24) and (25) , a n d as the edge is app roached , VDy —> b~(t)j2 T a k i n g i n to account the second d i f f r ac t i on t e r m a n d the geomet r i ca l opt i c s te rms as we l l as the step a n d s i gnum d i scont inu i t i e s i nd i ca te s t h a t as the edge is app roached , the s u m of the te rms matches the inc ident field, regardless of the app roach angle. For, e xamp le , w h e n b o t h geomet r i ca l opt i c s t e rms are + 1 , the d i f f r a c t i on te rms each app roach —0 .5 as a —»• 0. A l t e r n a t i v e l y , as the edge is app roached the s u m approaches (h, - V 2 + h2 - s2/2)S(t) = 6{t) ( ^ 1 - | + I ± * - f ) = 6{t) (36) for any source a n d receiver app roach angles 4>0 a n d <j>. C o m b i n i n g Vp^t) a n d Vc1(t) y ie lds the f o l l ow ing equa t i on . s i \ / a ( l + AAHit - a) 2n(t + aA\)\Jt — a = h1s(t + - A 1 ) - ^ I J J , V ^ . (37) 21 T h e second t e r m V2(t) may be eva luated in a s im i l a r fash ion to y ie ld yT[t) = "Vf^ («-?)/ C°S(^) . C°S(^) T[y/2Jt - R U + f c O S [ 0 - 0 O ] * + f c O s [ 0 + 0o] + M ( * + ^ 4 i ) ± M ( t + ^ A 2 ) . (38) In the above e q u a t i o n , the f irst t e r m is the d i f f rac ted f ie ld , the second t e r m is the d i rec t f ie ld a n d the t h i r d t e r m is the ref lected field. N o t e t h a t since the (plane wave) source is at i n f i n i t y , t i m e is measured f r o m the a r r i v a l of the p lane wave at the d i f f r ac t i ng edge a n d the d i r ec t a n d ref lected waves thus may a r r i ve at negat ive t imes . A ve r i f i c a t i on of the d i r ec t a n d ref lected a r r i v a l t imes of — ^ cos[0 ± 0 O ] m a y be deduced f r o m the p r o b l e m geometry . Re fe r r i n g to F i g u r e 4, denote the a r r i va l t imes of the d i f f rac ted , d i rec t a n d re f lected rays at po i n t A by t g , try and t^ respect ively. If the t i m e of i nc idence at the edge (point B) is ^ = 0, t hen ts = R/c = a , and , f r o m the d i a g r a m , z = Rsmcfi, z r s i n 0 7r r 2 = . , = . , > r3 = r j s m - - 2 0 o ) , sin 0o s m 0o 2 * r 3 - [ r 3 + # c o s ( 0 - 0 o ) ] , » , , q Q x tD = = — a c o s ( 0 — 0O1, a n d (39) c r-i - \rs, + # c o s ( 0 - 0o)l R s i n 0 s in 0 i2 . ^ = - = — - — : — — cos 20o cos (0 - 00) c c sm 0o c s m 0o c = —a [ co s ( 0 — 0 O ) — 2 s in 0 s i n 0 O ] = — a c o s ( 0 + 0 O ) . (40) T h u s t he geomet r i c a l opt i c s a r r i v a l t imes have been ver i f ied. F u r t h e r checks against o ther so lu t i ons w i l l be p e r f o r m e d i n C h a p t e r 3. 22 D i f f r a c t i o n response for an a r b i t r a r y source t ime f u n c t i o n can be eva l ua ted by con-v o l v i n g the source wavelet w i t h the above 6 f unc t i on response. T h i s w i l l be examined in C h a p t e r 4. 2.3 The line source problem R e f e r r i n g to F i gu re 3b, some of the s ymbo l s specif ic to th i s p r o b l e m are R = V P 2 +Po - 2pp0cos{(f> - c/>0) = yj{x- x0)2 + (y - y 0 ) 2 (41) R' = \jp2 +Pl- 2pp0 cos(c* + (j>0) = yj{x- x0)2 + (y + y 0 ) 2 (42) R R R1 Ri = p + po, a = — , 0! = - , 02 = — (43) c c c B e g i n w i t h a l ine source i nc iden t field V ( ( W ) = ' g » " ' " f t ) (44) T h e inverse Fou r i e r t r a n s f o r m of th i s is ^ ^ u j / i i i du Vt(t) = l R e r e - ^ M i TY Jo 4 i r f°° f°° = — / sin(wt) JQ(U0I) du — I cos(ut)Yo(udi) du 47T UO Jo (45) T h e l a t t e r integra l s , w h i c h are t a b u l a t e d i n E r d e l y i et a l . (1954, p.47 a n d p.99) a n d in G r a d s h t e y n and R y z h i k (1980, p.731), y i e l d v<it) = ^ r £ L . (46) 23 Figure 4: Geometrical construction for verifying the plane wave direct and reflected arrival times. 24 N o w cons ider the t o t a l field so lu t i on , der i ved by C l e m m o w (1950) f r o m a Sommer fe ld c on t ou r i n teg ra l rep resenta t ion by C a r s l a w (1899). I I r°° p%u rc V ' 2TT I y_mi Ju2 + 2u3, J--mi \/u2 -f )(3i J-m2 y/u2 + 2cj/32 VxHii^H, (47) where = sgn cos ( ^ - ^ ) ] \Ju(a — and m 2 = sgn [cos (^ j^ \Zw(a — /?2). We o r i g i na l l y a t t e m p t e d to sub s t i t u te in the in tegra l f o r m of the H a n k e l f u n c t i o n and p lay w i t h a t r i p l e i n teg ra l to do the I F T , b u t th i s came to naught. Instead, we p e r f o r m the inverse Fou r i e r t r a n s f o r m , us ing the no ta t i on s a n d procedures o u t l i n e d i n Sect ions 1.4 a n d 2.1, to o b t a i n • 2 1 r oo roo a x n VDl (t) = — R e / e---Vw* / du du (48) 27rJ Jo Jy/w(a-Pi) V r + 2wp! We let v = CJ + u2 j(2Pi) a n d reverse the order of i n teg ra t ion to o b t a i n ^ W = - i ^ R e / « d f , d t t . ( 4 9 ) 2TTV2/?I i o ^ V " We let p = (£ + /?i)/(2/3i), a n d t ake the rea l pa r t to get T/ ^ 1 r°° f T ^ T ^ t c o s p 2 - t ; t-/?! = 9-^2 / T T T / / ,2 7= d v dP- (50) 25 We let q = v\t- ft|, Lx = \t - ft|/[t + ft) and L2 = Lx(a + ft)/'(a - ft) to get 1 r°° f P ~ L VDl(t) = 7 ^ — / / 2 l 2 cos[p2 — sgn(i — ft)?] 2TT2J\t2 - ft21 ^ ^ i l V? dt; <ip 1 r 0 0 , TP L ===== / cosp / £ 2 _ / 3 2 i Jo y p 2 L l 2 /•P2i2 cos t7 dq dp 2TT2yJ\t2 - ft21 •'0 V i . sgn(£ - ft) r 0 0 . 2 [ P ~ L i sinq + f°°  /"p 2 sm<7 / smp / —— dq dp Jo Jp2Li Jq 2n2y/\t2 -ft21 J  V i i y/q 1 T r ^ J ^ - f t 2 ! {h + sgn(t - f t ) / 2 } , where (51) J, = / o °°cos(p 2 ) C j p ^ } - C {P^ /LT} dp, and / 2 = ^ sin(p2) [s j p y ^ } - 5 |P\/L7}] dp (52) (53) From Gradshteyn and Rhyzik (1980, p.654) or Erdelyi et al. (1954, p.42,98), r CO POO /rir / C(x)cos(b2x2)dx = S(x)sin(b2x2)dx = -^=-#(1 - 62) Jo Jo Ay/2b so that the integrals in Equation (51) take the form (54) J(A) s'mx dx r co roo \ C(Ax) cos x2 dx = / S(j4x) ii Jo Jo 1 f°° (v2 \ 1 f°° ( v2 \ A Jo C { v ) c O S { A 2 ) d V = A J o S ^ ^ [ ^ ) d v  4 7 ^ \ ^ Ay/2 H(A2 - 1) (55) 26 h = h = J{y/L2) - J ( V ' L 1 ) = { i ? ( L 2 - 1) - H(L, - 1)} (56) Since Ij + s gn ( i — (i\)I2 = hH(t — ft), we on ly need t o cons ider Z > ft a n d can drop the abso lu te value signs in L j and L2 to o b t a i n * - ft -2/?! , , 1 [(« - f t ) ( a + ft)-(* + ft)(c* - f t ) ] ( i + ft)(a-ft) + ft* - a f t - ft2 - at + ft* + ft2 - a f t _ 2ft(i - a) (* + ft)(a-ft) (* + ft)(a-ft)' S ince a > ft, i7 ( A 2 - 1) = # ( * - a ) , and WO 7) + sgn(r - ft)J2 = x/2ff(t - ft) ^  s/2^H(t- fa)H(t - a) H{t-a) 4 v/27rv/?v/t2 - ft2 47^ - ft2 2 7 T 0 2 - ft2 (59) (60) V G l (0 = 2 l i m (0 = H{-tP}±_ (61) / x 2h1H(t - ^) - SiH(t - ^) , , T h e second t e r m V ^ i ) may be eva lua ted i n a s im i l a r f a sh ion t o y i e l d V (t) .= 2 k l H { t ' * } " S l H { t " ^ J_ 2 k 2 H { t ~ ^ ~ S i H { t ~ ^ (63) 27 We now examine the far-field approximation to the exact frequency domain one. Since this assumes w a » 1, (or Rx > A/(27r)), the portion of the I F T /0°° du for frequencies near zero is distorted. However, assuming that a is large enough to minimize the effects of this distortion, the I F T should produce an approximate far field time domain solution. Alternatively, convolution w i t h a realistic wavelet would filter out this low frequency distortion. For this case, the first term in the frequency domain far field diffraction solution, denoted by V^(u), is (Bowman et al. 1969) (64) The inverse Fourier transform of the above is (0 2V2TT i I Jo ( f°° cos(u2 )C t-0i du H(t - a) (65) 47ry/{a + 01){t-01)' The geometrical optics portion of the far field solution is H(t-Bx) (66) E q u a t i o n (65) implies that 28 V £ ( Q = E q u a t i o n (65) = /1 + ft V D l (t) E q u a t i o n (60) V « + ft T h u s ( i) w o u l d be v a l i d on ly for t close to a, a l t hough the degree of error decreases as a increases, i.e. i n the far f ie ld . T h i s makes sense in te rms of an expans ion of the exact ope r a t o r near the onset t i m e t = a. If t — a + e, where e <C a, t hen ' l - n , . 0 , ) (68) \A2 - ft2 Vt^TiV*~+Ti^ + ^ ^ ( a + ftHt-ft) V 2(a + ft) T h e r a t i o of the two sides of the above equa t i on verif ies E q u a t i o n (67) a n d the fore-go ing i n t u i t i v e d i scuss ion. 2.4 The Point Source Problem. Re fe r r i n g to F i g u r e 3(c), some n o t a t i o n spec i f ic t o th i s p r o b l e m is R = \Jp2 + p i - 2pp0 cos(0 - 0O) + (z - so ) 2 = XQ)2 + (y - y0)2 + (z - z0)2 R' = \ J P 2 + PO- 2pp0 cos(0 + <t>0) + (z- zQ)2 = y/(x- x0)2 + (y + t/o)2 + (z - 20)2 *2i = r + r 0 = VWPO)2 + (*-*O)2 (69) A l s o let a = Ri/c, fix — R/c a n d ft = R'/c. T h e po i n t source inc ident field m a y be represnted as V M = ^ « V ( W = ^  (70) 29 a n d the re su l tan t t o t a l field, der i ved by Ca r s l aw (1899) us ing a Sommer fe l d contour i n teg ra l rep re senta t i on , is VT{u) I. °° UH[1]{U2 + wf t ) , *.±ir ^ ! y + "W ,» y/u2 + 2 w f t C J-m> y/U2 + 2u02 Vx(u) ± V2(u) (71) where m = sgn (cos [ ^ ] ) y/w[a. - ft) and m' = sgn (cos [ ^ ] ) \Ju{a-02). W e now do the inverse Fou r i e r t r an s f o rm, us ing the no ta t i on s a n d procedures ou t l i n ed i n Sect ions 1.4 a n d 2.1, to o b t a i n 1 _ r° - R e / 7 T Jo c " , w * V i ) l (w) du; 1 r°° - — Im/ e- , w l 7 T C JO J rco /-co rC Jo Jy/u>(a-P} du du r ~ u >g {x ) ( t t 2 + c j jg ) Jy/ui(a-0) yju2 + 2w/3 w [ s i n ( w i ) f t ( u 2 + w/?) - c o s ( w t ) Y i ( u 2 + w/?)] V ' u 2 + 2w/? (72) du du; We now let v = 0 + u 2 /w to o b t a i n 1 A00 r 27TC JO Ja 0 0 f°° Wy/u [sm(ut)Ji(uv) - cos(u>t)Yi(wv)] 0J\ v l r 0 0 — / S 1 ITVC JO s i n ut I3 duj lire JO ! - f t 2 COS Ut I4 du. dv du (73) 30 I3 a n d 14 m a y be in teg ra ted by pa r t s as fo l lows. /•CO '3 = / (v2 - 0l)-i d\J0{uv)} J a \A2 - 3\ -J0(uja) \J<*2 " ft2 V=°° + 1 [°° (2 (2«)J0(q;t;) ( w 2 _ ^ 2 ) 3 / 2 + /•CO UJO(CJU) ( u 2 - diyi-dv (74) /•CO /4 = / (^-^j-sdiroM] •/a YQ{uv) \v=°° 1 y°° (2u)y 0(m;) ^ T T ^ J L=a + 2 7a ( „ 2 - f t 2 ) 3 / 2 du - y 0 ( f a ; a ) r°° vY0{iov) dv (75) In p e r f o r m i n g the u i n tegra l we w i l l need integra l s t a b u l a t e d i n E r d e l y i et a l . (1954, p.47,99) a n d i n G r a d s h t e y n a n d R h y z i k (1980, p.731) as j(K) = s\n(ut)J0{uK) du = - cos[ut)Y0(ua) du = ^ ~ ^ (76) 31 E q u a t i o n s (73) t h r ough (76) i m p l y t h a t v ».«> = 2T.X s'm(ut)Jo(ua) dw 1 r°° r°° vs'm(ut)JQ(u}v) 1 r°° f 2TTC Ja JO 1 7TC 1 7TC . . — dw + - — yja2 - ft2 2 7 r c du /•OO /"C Jo JO \/«2 - ft 1 Z - 0 0 cos(w<) Y0(WQ:) 27TC io j 'M _ f°° vj{v) #(* - a) /•«> vH(t - v) iC dw dv 5 i > c o s ( w £ ) Y o ( w u ) a 2 - f t 2 ) ( r 2 - a 2 ) ^ ( u 2 - / ? 2 ) ! ^ * ^ ^ du dw dv (77) Deno te the second t e r m above by T2. If t < a, T2 = 0, b r i n g i n g i n a step f u n c t i o n H(t — a. A l s o note t h a t the in f in i te l i m i t is rep laced by t because of the H(t — v) in the i n t eg r and , and make the t r a n s f o r m a t i o n u — v2 — 0\ to o b t a i n H(t-a) [*2-f>\ 2-KC r^ Pl du '"'-fi UyJU{t2 -fil-U H{t-a) t * - f i -*c(t2 - ft2) H{t-a) t2 c r 7TC{t2 - ft2) V OC2 - ft2 (78) Hence, f r o m E q u a t i o n s (77) and (78), VDl (t) H{t - a) or H{t-a)yJc?-Bl vcyJcJ^Pl I yf*^2 t2 - ft j 7rc(r2 - (32) y / ^ o ? (79) 32 The a J f l VDi[t) is needed for the geometrical optics term VGl(t). This l imit is zero except when t = a, when it is infinite. This seems to be like a delta function. We now check this by examining its behaviour as an integrand. Since a and ft are constants, interchanging the order of l imit and integration is val id. U = hm ot->Pi J - C O dt = l i m /oo -oo H(t - a) Jo? - f>\ 01 J -l i m — l i m — lim a—Pi 7 r c ( i 2 -df)\/t2 - a2 \/«2 - 01 dt 7TC r dt [t2 - 3\)sjt2 - a2 23 — i f 17TC [Ja dt r J a dt \l*2 - 31 23  Pi f r°° l7TC XJia-P^vfv {t-3i)Vt2~:ro7 Ja [t + 31)^/^0? dv yj 2 + 28v + [31 - a 2 ) /•°° du 1 vyjv2 - 23v + (ft 2 - a2) > (80) 33 These integrals are listed in Gradshteyn and Rhyzik (1980, p.84). I2 = — lim <, sm 2f t7TC ft2 - a 2 + ft; av sm a-0 PI - a 2 - pv av a+P 1 , , • -1 A • -1 -Pi l im sm sm 2ft7rc a-*Pi { a \ a . P' - a 2 + p 1 a - p - \ . P l - J - P x a - P l sm I - + sin -a2 — fta a2 + fta \ - lim | 2 s i n - 1 f — ) - s i n - ^ - l ) + s i n - 1 ( - l ) l 2p lim -KR a—PI sm \ Ct 1 7T _ 1 ^ R 2 ~~ 2R (81) As a second check, we test the picking property of the delta function for the simple function f(t) = (t + ft)\A + a, making the successive transformations t = u + a and u = (a — Pi) t a n 2 0, as follows. f(t) lim V i , , (0 dt=— lim V"2 - ft2 / R , n -00 a->Pi 7TC a->Pi v J a (t — Pijy/t — a 2 f t /-^ 7^ /*°° du 2 f t 2 r\ = hm \Ja2 - Pf / —=•. — = hm yja2 - P2 , / TT or-,8, V ^ y 0 y/ttfu + a - Pi] 7T a - f t V - ft 7o = 2 f t lim v/a + ft = 2 f t ^ 2 f t = / ( f t ) d0 (82) Equations (79), (81) and (82) imply that VG l(t) = 2 lim VDl(t) = 6 { t J ' \ and (83) 34 V1(t) = hVGl[t) - sVDl[t) H cos <f> - (f>0 S(t - ft) ^ a2 ~ ftsgn (cos H(t - a) R 7Tc{t2 -82)Vt2 - a2 (84) T h e second t e r m V2(t) m a y be eva luated i n a s im i l a r fa sh ion to y i e l d VT(t) = V1(t)±V2(t) = H cos 4> - 4>o R ± H cos 4> + <t>o ( t - ^ ) r s g n (cos [ * ^ ] ) ^JRJ^R2 c2t2 - R? ± sgn (cos [*±*u]) ^ - ( j ? ) c2t2 - [R')2 (85) In t he above equat i on the first t e r m is the d i rec t a r r i v a l , the second t e r m is the ref lected a r r i v a l and the last two te rms the d i f f r a c t i on a r r i v a l . Ins tead of the exact s o l u t i on , we now exam ine the far f ie ld s o l u t i on , ( B o w m a n et a l . , 1969) TTc2a(a + ft) "*/ 4>F + \Ma-ft) (86) a n d eva luate , w i t h re s t r i c t ions s im i l a r to those for the l ine source case,the inverse Fou r i e r t r a n s f o r m D l V J TT]/ 7TC2a(o + ft) [Jo Jy/ula-h) (87) 35 P r o c e e d i n g as for the p lane wave p r o b l e m , 7r V 7Tc-a(a + ft) Jo Jo (88) y/a - faH(t - a) _ H(t-a) / a - ft 7TC2a{a + ft) 2yfjv(t - ft)^A - a ncy/t - a(t - ft) V 2a(a + ft) VpM = s/tTa~(t + ft) = ft + pA t + a { 8 Q ) T h u s V£i (t) is v a l i d on ly for a short t i m e after t = a, a l t hough the w i n d o w of accu racy increases as a increases, i.e. in the far f ie ld . We now examine th is i n terms of an expan s i on of Vz),(f) near the onset t i m e t — a, as for the l ine source. If t — a + e, where e <C a, t h e n 1 ( « ' - P t ) ( * - P)VT^(a + ft) (l + ^ ) V ^ v ^ T ^ V/ 2 a ^ T ^ ( a + ft)(*-ft) V 4 a 7 \ <* + Pi) (!-^±m. (90) ^/2a(i - a){a + P1)(t - ft) V 4a ( a + ft)/' T h e r a t i o of the sides of the above equa t i on verif ies E q u a t i o n (89) a n d the above i n t u i t i v e d i scus s ion of the w i n d o w of va l i d i t y . 36 2.5 Dipole Sources. T h e above po i n t source so lut ions m a y be ex tended to the t i m e d o m a i n so lu t i on for d i f f r a c t i on of an e lect romagnet ic d ipo le field by a per fect l y c o n d u c t i n g ha l f p lane. For a n e lec t r i c d ipo le at (po, <f>o, zQ) w i t h a momen t (47re:)c equ iva lent to an e lect r ic H e r t z vec to r ri* = ceiuPl/R, where c = x s in 0 cos $ + y s in 0 s in $ + z cos (91) a rep re sen ta t i on of the t o t a l field is ( B o w m a n et a l . 1969) rie(w) = x + y lVf(u) mVj (u) — MQ '{ua)[l sm m c o s — ) s m — PPo 2 2 2 0(K • <t> m cos — ) sm — 2 ; 2 1 ITW( W ; • ^ ° ,—Ho M ( | s m v cJppo 2 (92) znVT (OJ). In the above equa t i on , V^(to) and V^t(oj) are the soft and r i g i d acoust ic po i n t source so lu t ions g iven in E q u a t i o n (85) and / = s i n © cos $ , m = s i n © s i n $ a n d n = c o s © are the d i r e c t i on cosines of the d i po l e vector c.The inverse Fou r i e r t r an s fo rms Vf{t) a n d Vf (t) have been g iven i n E q u a t i o n (85), a n d the t r an s f o rms of the other te rms are eas i ly eva luated, us ing E q u a t i o n (76) to y i e l d TLe(t) = x[lVf(t) + (lo-i - mo2)S(t)} + y [ m V r + ( * ) - (la3 - mo4)S(t)] + z n V f ( i ) , (93) where ox = s i n ( ^ 0 /2 ) sin((£/2), o2 = cos(</>0/2) sin(^>/2), cr3 = sin(<£ 0/2) cos (^/2) , cr4 = cos(r/>o/2) cos(<ji>/2) a n d 2 / si (cos ) + s * ( cos 4>+'P<> 2 ) "\ H(t- a) 7TC a ' (94) 37 G i v e n the above a n d the fact t h a t the IFT and the operators V x a n d commute , it is poss ib le t o ca l cu l a te the e lectr ic a n d magne t i c d ipo les v i a Q E ( t ) = V x V x n e(t) a nd H ( i ) = e V x ^ ne(*). (95) If i n s tead the inc ident f ie ld is a magnet i c d ipo le w i t h momen t 4 7 r c , t hen d E ( < ) = V x V x n m (r) and H{t) = E V X ^n m ( t ) , (96) where , f r o m B o w m a n et a l . (1969), n m (r) = x[lV^{t) + {lo4 + mo3)S{t)]+y[mVf{t) + [lo2 + mo^Sty)} + znV+( / j ) . ( 9 7 ) N o t e t h a t a v e r t i c a l (c = z ) e lectr ic d ipo le s o l u t i o n Ile(*) reduces to iV^ (t) a n d a v e r t i c a l magne t i c d i po l e s o l u t i on TLm(i) reduces t o zVj(t). A l s o note t h a t for the V x ope ra t i on s , e i ther the quant i t i e s def ined i n t e rms of p, <j> and z mus t be redef ined i n te rms of x, y a n d z, or the u n i t vectors x a n d y mus t be t r an s f o rmed to p a n d </>. D u e to con s t r a i n t s on t i m e and a desire to keep the e q u a t i o n number s below (200), deta i l s are left for a fu tu re p u b l i c a t i o n . A l s o , before p roceed ing to i m p l e m e n t a t i o n s s i m u l a t i n g d i r e c t i o n a l m a r i n e se i smic sources, we have t o do more research into acoust i c d ipoles a n d t he i r r e l a t i on sh i p to the e lec t romagnet i c ones. Q u a l i t a t i v e l y , d i f f e ren t i a t i on of the po i n t source so lu t ions p roduces d ipo le t e rms . 38 3 . COMPARISON WITH OTHER SOLUTIONS. 3.1 Plane wave solution comparisons. A s a check we c o m p a r e d the p lane wave s o l u t i on , E q u a t i o n (38), to the so lu t i on for S H d i f f r a c t i on g iven by de H o o p (1958 pp.37-41), no t i ng t ha t h i s 6S — n — <fio and t h a t he used d i sp l acement p o t e n t i a l i n s tead of ve loc i ty p o t e n t i a l , so t h a t his r i g i d ha l f p l ane s o l u t i on cor responds to ou r weak one and v ice versa. W i t h th i s t a k e n into account the so lu t ions are i den t i ca l . A l s o , we c o m p a r e d ou r s o l u t i on w i t h a l i m i t i n g f o r m of the wedge so l u t i on e xpan -s ion g i ven by Felsen a n d M a r c u v i t z (1973). T h e first d i f f rac t ion t e r m of the i r s o l u t i on reduces to 1 H(t - a) (98) 7T y/t2 - a2 where /? = cosh 1(t/a) a n d (99) U s i n g the above def in i t ions and ha l f -ang le f o rmu lae , the i r s o l u t i on becomes - V ^ ( c o s [ ^ ] ) H{t - a) (100) TT y/t2 - a2 £ + C O s ( 0 - 0 o ) 7T y/2-Jt - a[t + a + cos(0 - <p0)}' 39 in agreement w i t h E q u a t i o n (38). A n u m e r i c a l check of our inverse Fou r i e r t r a n s f o r m c a l c u l a t i o n is dep i c ted in F i g -ure 5. T h e Fresne l in tegra l c a l cu l a ted in the f requency d o m a i n is Fcaic(u>) and its d i g i t a l inverse Fou r i e r t r a n s f o r m is F i r a n s ( Z ) - T h e ana l y t i c Fou r i e r t r a n s f o r m c a l -c u l a t e d d i r ec t l y in the t ime d o m a i n is Fcaic{t) and its d i g i t a l Four ie r t r an s f o rm is Ftrans{u). C on s i de r i n g the effects of finite d a t a l eng th , non - i n f i n i t e s ima l s a m p l i n g i n -t e r va l a n d the s ingu lar i ty , Ftrans(to) agrees we l l w i t h Fcaic(oj) and Ftrans(t) agrees we l l w i t h Fcaic(t), w i t h on ly s l ight, s m o o t h i n g of the s ingu lar i ty . See Sect ion 3.3.1 for f u r t he r c o r r o b o r a t i o n w i t h the l i m i t of the po in t source so lu -t i o n as the source is removed t o in f in i ty . 3.2 Line source solution verification. T h e l ine source s o l u t i on , E q u a t i o n (63), matches t h a t der i ved by Felsen a n d M a r c u v i t z (1973, p.672). A l s o , the geomet r i ca l op t i c s te rms (61) agree w i t h those i n a d i scuss ion of the r a d i a t i o n p a t t e r n of a b u r i e d l ine source b y G i l b e r t a n d K n o p o f f (1961, p.629, equat i on s 27,28), and the fa r - f ie ld f o r m of these te rms (66) agrees w i t h the far field f o r m de r i ved by He lmbe rge r et a l . (1986) f r o m G i l b e r t and K n o p o f f ' s s o l u t i on . 40 0 40 80 120 freq (Hz) trans' 0 40 80 120 freq (Hz) (<*>) Fcalctt) 5 4 I 3 0 2 1 0 2 3 4 t (sec) Figure 5: N u m e r i c a l check of the I F T of the F resne l i n teg ra l : (a) T h e Fresne l i n -teg ra l c a l c u l a t e d i n the f requency d o m a i n , (b) T h e d i g i t a l Fou r i e r t r a n s f o r m of (d). (c) T h e d i g i t a l inverse Fou r i e r t r a n s f o r m of (a), (d) T h e a n a l y t i c a l l y c a l c u l a t e d inverse Four ie r t r a n s f o r m of (a). In each s ubp l o t , t he y -ax i s l abe l amp denotes the amp l i t ude of the p l o t t e d f u n c t i o n . 41 3.3 Point source comparisons. 3.3.1 Verification in the plane wave limit. An obvious solution check involves taking the limit of the point source diffraction term (with geometrical spreading removed) as the source moves to infinity, and redefining the time origin for comparison with the plane wave solution, as follows. We now examine the first term of the point source diffraction operator, (79). For simplicity, assume that z — z0 = 0 and make the substitutions n = p/c, r)Q — po/c, and ap = a — n0 = rj, to obtain vDi(t) H(t-a)y/a*-82 ncVt2 - a2[t2 - 02) H{tp - ap)^2ri0ap{l + Ax) 7rc^2r)0(tp - ap) + {t2p - a2p){2n0[tp + a^} t2 (101) Now multiply by the factor po — cr)o to remove the geometrical spreading factor associated with the point source, and take the limit as r)0 —> oo to simulate a plane 42 wave, as fo l lows. L = l i m (ri0c)VDl(t) T7o—>oo lim In—>oo H{tp - ap)noyj2rj0ap(l + Ax) *yj2r,0(tp - ap) + [t\ - aj) (2r,Q{tp + apA,} + {tj - a*}) \Japi1 + ^ 1) ,. 1 h m 2n r j o - » o o (102) 2iYy/tp - ap(tp + a p A i ) ' i n agreement w i t h E q u a t i o n (34). 3.3.2 Comparison with Wait's step function solution. T h e exact so lu t i on (85), wh i ch has no res t r i c t ions o n the angles <fi a n d <j>o> >s f ° r a d e l t a f u n c t i o n source e x c i t a t i o n a n d may thus be used to generate (by convo lu t i on ) the response for an a r b i t r a r y source t i m e f unc t i on . E i t h e r c onvo l v i n g o u r so l u t i on w i t h a step f u n c t i o n or d i f fe rent ia t ing W a i t ' s (1957) s o l u t i on for a step f u n c t i o n e x c i t a t i o n w i t h respect to t ime shou ld ver i f y ou r so lu t i on . W a i t ' s (1957) s o l u t i on for a source 43 excitation pQH(t) / (4TYR) , converted to our variables, is 9w[t) Po 4n H{t-R/c) H(t-R'/c) R Po R' 8n2R 7T , JR{ ~ R2Ct tan"" 1 v 2 R\/{ct)2 - Rj H(t- Ri/c) Po 8n2R' 7T , \/Rl-(R')2ct\ 2 I - - t a n " 1 V Hlt-RJc) 2 R^(cty-Rl)\ -Po 4n2R H(t - Rjc) tan — — ± tan ct \/Rl - R2 ctyjR2 - {R'Y + Po 4TX H(t-R/c) H(t-R'/c) R ± R' (103) Now consider the convolution of our solution with a step function source t ime function 9m{t) = VT(t) p0H(t) _ po r°° 47T 47T.J-f°° H{t-u)VT(u)du = — f VT(u)du (104) J-oo 4TX J-OO where VT(i) is given in Equat ion (85). Note that r 6(u- d ) d u = | x ' J-°° [ o, t > 0 t < 0 = H(t - 0) and (105) rt 0 , t < a I rt / H{u - a)<j>{u) du= I \ = H{t- a) </>{u) du { /^(u)du, t > a 1 (106) 44 These last two equat ions i m p l y t ha t 9m[t) Po 47T H[t - R/c) H(t - R'/c) R PoHjt - Rx/c) 4n2c R' \JR\2R du ± 4 - s R2, (107) F r o m E q u a t i o n (80) a n d f r o m G r a d s h t e y n a n d R h y z i k (1980, p.84), the integra l dv a [v2-f32)y/v2 - a2 t a n i [ §v •s/a2-02\/v2-a2 t a n " -1 / 0v+a2 y/a2-p2s/v2-a2 2/3y/a2 - {32 t a n " 1 / /J t-a 2 yJa2-P2s/t2-a2 + t an - l Jt+a2 yJa2-&2-Jt2-a2 2(3y/CL2 ~ P2 (108) T h i s resu l t leads to the fo l l ow ing f o r m of gm. 9m{t) Po_ 4n H{t-R/c) H{t-R'/c) R R' Po 8w2R t a n Ret - R\ + t a n ^R\ - R2\J(ct)2 - RI Ret + R\ y]Rl-R*yl{cty-R\) T a s im i l a r t e r m w i t h R' i n s tead of R. H[t - Rx/c) (109) 45 C o m p a r i n g gm(t) a n d gw(t), they are i den t i ca l if t a n - l Ret- Rl y/Rl - R^{ctf - R\ t a n " 2 t a n ' Ret + R\ Rl - R2yJ{cty - Rl a (Ry/jcty-RJ \JR\ - R2ct (110) L e t the above equa t i on be t a n 1 Zi + t a n 1 Z2 = 2 t a n 1 Z$. T h e n , u s ing the tangent a d d i t i o n f o r m u l a , 2cRt t a n taiT^Zi) + t a n _ 1 ( Z 2 ) 1 - ZXZ2 i R2(ct)2-R* (R2-R2)\(cty--R2} 2cRtx/R21 - R2\J{ct)2 - R2 [Rl - R2)\{ct)2 - Rl] - R2(ct)2 + Rf 2ctRy/Rl - R2\](ct)2 - Rl [Rl - R2){ct)2 - R2[{ct)2 - Rl] (111) a n d tan[2tan _ 1 (Z 3 ) ] 2Ry/{ct)*-R\ 2Z3 _ y/RJ-R2Ct 1 - z\ _ R2i(cty-R*} {R2-R2)(cty-2ctR\jR2 - R2x/{ct)2 - R2 {Rl - R2){ct)2 - R2[{ct)2 - Rf (112) T h e y are i den t i c a l , a n d thus our s o l u t i on has been con f i rmed . A l t e r n a t e l y , note t ha t s ince 46 (113) dgw dt d_ dt Po_ 47T H{t) * VT{t) ^8{t)*VT{t). 4ix (114) Thus , our solution may be checked (in the region of validity of Wait 's solution) by taking the derivative of Wait's solution, as follows: 4*9* Po H i t - f i x ) ±H[t-fo) R R' Hjt - a) nR tan" ± tan V«2 - PI * v 7 " 2 - P (115) 47T dgw Po dt S i t - p i ) ±6it-p2) R R' tan 6jt - a) TTR Hjt -a) d nR dt OL tan ty/cJ-J2 ± tan .iPisf* or tsjct2 - P? 2 J t \ / « 2 - Pi ± tan 2 _ a2 V"2 - ^ 2 2 J (116) The t h i r d term may be neglected, since when t = a the tan 1 term disappears. Comparing the above to V r ( / j ) the first two terms match perfectly and the last w i l l if d DG = — t a n - 1 dt P^Jt2 - a2 /Va2 - P2 Jt2 - a2(r2 - p2)' (117) 47 S e t t i n g 7 = 0j\J a2 — 02, the above equat i on becomes Da = dt tan 7 a a* 2t /3a 2 v 7 ^ a ' [t2 + 7 2 ( * 2 - «2)] s/a*-0*VF=tf , 2 , /3 2(t 2-a 2)  1 ~r a2-/?2 /3a 2 V « 2 - /32 /3\/a 2 - /32 v 7 * 2 - a 2 [ a 2 * 2 - /?2*2 + /3 2 * 2 - a 2/3 2] ( i 2 -/3 2)xA 2 - a 2 ' (118) E q u a t i o n s 111, 112 a n d 118 thus ver i f y our s o l u t i on w i t h Wa i t ' s . 3 . 3 . 3 Compar i son w i t h K i r chho f f theory. We now compa re ou r d i f f r a c t i on ope ra to r w i t h the one de r i ved by T ro rey (1970) and B e r r y h i l l (1977) u s ing the K i r c h h o f f a p p r o x i m a t i o n . B e r r y h i l l ' s ope ra to r conver ted to ou r n o t a t i o n a n d w i t h his n o r m a l i z a t i o n fac to r removed (by m u l t i p l y i n g by Rx) is Do 02 x/oc* - 0: t [7Tc{t2 -0l)\/t2 - a2 J i n s t ead of our zero-offset ope ra to r for the same geometry, (119) DT = a + 7rct2y/t2 - a2 ?rc{t2 - 01)Vt2 - a 2 ' (120) The re f o re his d i f f r ac t i on ope ra to r is j u s t 02/t t imes the second t e r m of ours. T h i s w i l l be accep tab le on l y for t races near or at the apex of the h y p e r b o l a ( a > /32) a n d t imes near the onset t ime (t > a). T h e f irst t e r m w i l l d i s to r t the results s l i gh t l y even nea r these t imes . A l s o , for l a ter t imes a n d for t races on the flanks of the h y p e r b o l a 48 a m p l i t u d e amp l i t ude s and phases w i l l be d i s t o r ted . T h i s , coup led w i t h a d i s to r t i on of the phase reversa l at the apex, shou ld affect d i f f r ac t i on s tack ing , co r re l a t i on or m i g r a t i o n techn iques. T h e second t e r m of our ope ra to r w i l l f l i p i n s ign as the apex of the h y p e r b o l a is passed, w h i l e the f i r s t t e r m w i l l r e m a i n negat ive th roughout . Thu s , w h i l e Be r r yh i l P s o p e r a t o r is s y m m e t r i c about the apex, ours exh ib i t s s t ronger amp l i t ude s on the flank of the h y p e r b o l a be low the ref lector. A l s o , a perfect 180 deg phase shift is no t observed at thesapex. T h i s is d i scussed f u r the r in C h a p t e r 5. N o t e t h a t many of H i l t e r m a n n ' s (1970) e x p e r i m e n t a l results p r oduced s y m m e t r i c a l h y b e r b o l a e s ince the d i f f r a c t i n g bod ies were s y m m e t r i c . E v e n for these there are some a m p l i t u d e p rob l ems for his K i r c h h o f f so lut ions on the f lanks of the h y p e r b o l a . M o d e l s w h i c h were not s y m m e t r i c d i d not e x h i b i t near ly as good a m a t c h . A l s o note t h a t ou r non-zero offset s o l u t i on is m u c h s imp le r in f o r m t h a n those de r i ved by B e r r y h i l l (1977) a n d T ro rey (1977). T h i s w i l l f a c i l i t a te c a l c u l a t i o n of the errors i nvo l ved i n C D P s t a ck i n g of d i f f rac t ions and i n var ious d i f f r ac t i on process ing techn iques . 3.3.4 Comparison with wedge solutions. F i n a l l y , before p roceed ing t o examples , we compa re the ha l f p lane po in t source d i f f rac-t i o n s o l u t i o n w i t h l i m i t i n g f o rms of wedge so lut ions. We now examine the wedge so-l u t i o n de r i ved in Felsen a n d M a r c u v i t z (1973), w h i c h mus t be m u l t i p l i e d by a factor of 47r for t he i r source t o m a t c h ours. W h e n the wedge angle approaches 27r the i r geomet r i c a l opt ics t e rms agree w i t h (85) and the the first d i f f r ac t i on t e r m reduces to 49 If %p = (0-</3o)/2, then 2pp0/c2 = ( a 2 - ft2)/(l + cos 0) and W i t h the half angle formulae and the further definition ReLB(0,0 o;*p) = r-^- ;—, coshp + cos^y their solution becomes cH{t - a) \ cos(V>/2) cosh(/?/2) { /9/>o sinh /? [ cos ip + cosh /3 j SjHjt - q ) ( l + c o s ^ ) 1 5 ( a 2 - ft2)15 c(« 2 - / 3 2 ) 1 5 { ( « 2 - ft2) cos V + t2 - ft2 + [ t 2 - a 2 ) cos V} J ( l + cos x P ) [ t 2 - a 2 ) + 2(a 2 - ft2) x — ^ { t 2 ( l + cos^) - ft2 - a 2 c o s V } 2 - (a 2 - ft2)2 s,#(* - a)( l + cosi>) 1 5\Jct 2 - ft^l + cosV>)(r2 - a2) + 2(a2 - ft2) c(i 2 - ft2)\zV(l + cost/;) - ft2 - a2 cost/3]2 - (a 2 - ft2)2 R.c[*-R) ' W H E R E 50 11 _ a2t2 _ a2 (_a2 \cQ*i>-l]) _ a2 ( ~2^ \ 1 \ [cosr / i+lj/ \ 1+cos ip J t2 - a 2 + 2{a2 - ft)/(l +cosV) = \/t2 - a 2 . (126) S u b s t i t u t i o n of (126) into (125) y ie ld s • SlH(t-aya2-02 (127) cVi2 - ct2(t2-02) T h i s s o l u t i on , on compa r i s on to E q u a t i o n (85), is i ncor rec t by a f a c to r of 7T. A t f i r s t , in a m o m e n t of pan i c , we env i saged the pos s ib i l i t y t ha t our s o l u t i on was incorrect . Howeve r , on cons ide ra t i on of the two ear l ie r i ndependent ve r i f i ca t ions th ings looked b r i gh te r . T h u s , i n o rder to avo id a rgument s of c i r cu l a r reason ing a n d t o break the t ie of so lu t ions agreeing a n d d i sagree ing, we compa red our s o l u t i o n t o a l i m i t i n g f o r m of the po in t source wedge so l u t i on de r i ved by B i o t and To l s toy (1957) a n d later s imp l i f i e d by Jebsen and M e d w i n (1982) a n d H u t t o n (1987). F o r a 8 f u n c t i o n source a n d a wedge angle of 27r, the i r s o l u t i on becomes (us ing n o t a t i o n def ined above for the 51 Felsen and M a r c u v i t z po i n t source so lu t i on ) , cH(t-a) f cos (^/2) cos(0/2) 4npposinh0 \cosh(/3/2) - s in (^/2) + cosh(/?/2) + s i n ( ^ / 2 ) cH(t-a) f 2 cos(0/2) cosh(/?/2) 1 4 7 r p p 0 s inh/? [ cosh 2(/3/2) -sin2(0/-2) j cH(t-a) \ cos(^/2)cosh(/?/2) 2nppo s inh 0 { ( l + cosh/?)/2 - ( l - cos V )/2 ccos(r/>/2) cosh(/3/2) 7rpp 0 s inh /?(cos + cosh 0) H{t - a), (128) w h i c h is so lu t i on (124) m u l t i p l i e d by a factor of l /V , i n agreement w i t h our s o l u t i on . 52 4 . P R A C T I C A L A P P L I C A T I O N . 4.1 Overview. In the r ema inde r of the thesis, the f ac to r H(t — a) is o m i t t e d , i.e., t he d i f f rac t ion response is c a l c u l a t e d for t > a only. T h e above p lane wave and po i n t source impu l se responses mus t be convo l ved w i t h a source wavelet t o g iven the response for a real i s t ic source. T h e geomet r i ca l opt i c s te rms are eas i ly eva luated. G i v e n an ana l y t i c wavelet, i t m i gh t be poss ib le to eva luate the convo l ved d i f f r ac t i on t e r m d i rect ly . However , i n p r a c t i c e the c onvo l u t i o n in tegra l mus t be a p p r o x i m a t e d by a d iscrete a n d t r u n c a t e d s u m . T w o p r ob l ems are assoc iated w i t h th i s : one of c onvo l v i n g n u m e r i c a l l y w i t h a s i n gu l a r i t y a n d one of r emov i ng a r t i fac t s p r oduced by t r u n c a t i o n . O f t e n the first has been hand l ed by s t a r t i n g the convo l u t i on at t = a + At, not at t = a. T h i s results i n some a m p l i t u d e deg radat ion as we sha l l demons t r a te later in the examp le s sect ion. B e r r y h h i l l (1979) hand l ed the second p r o b l e m severa l by an unnecessar i l y comp lex scheme w h i c h e x p l o i t e d the sepa ra t i on between the d i f f r ac t i on wavelet a n d the t r u n -c a t i o n ghost a n d cou ld not hand le cases w h e n the two inter fered. R e l a t e d prob lems i n v o l v i n g t r u n c a t e d integra ls were t a c k l e d by T h o m s o n a n d C h a p m a n (1986). G i v e n a wave let W(t) of w i d t h tw (def ined on 0 < t < tw) a n d s a m p l i n g in terva l A * a n d the p lane wave or po i n t source impu l se d i f f r ac t i on response D(t) the convo lved 53 d i f f r a c t i on response is rMm(t,TD) Q(t) = W(t)*D{t)= W{t -u)D{u)du (129) J M a x ( a . / - ( , „ ) Fo r t > a + tw the convo l u t i on m a y be eva luated i n the n o r m a l fa sh ion , bu t for t < a + tw the p r o b l e m of convo l v i n g w i t h a s ingu la r i t y is encounte red , due to the inverse square root s i ngu la r i t y at the lower l i m i t in a l l forms of D(u). A l s o , if the ope r a t o r is t r u n c a t e d at To the p r o b l e m of t r u n c a t i o n artefacts is encountered. T h e nex t sect ions dea l i ng w i t h these p rob l ems w i l l focus on the f irst t e r m of the po in t source d i f f r ac t i on ope ra to r , w h i c h f r o m E q u a t i o n (79) w i l l t ake the f o r m s j a 2 - 02 DU) = V , , (130) 4.2 Hand l i ng the singularity. Fo r now, we assume the t r u n a c t i o n co r rec t i on is h a n d l e d separately, if needed, so t ha t we c a n concent ra te on the s i ngu l a r i t y co r rec t i on . If the ope ra to r is at least as long as the wavelet the two cor rec t ions are never needed s imu l taneous l y , a n d for shor ter ope r a t o r lengths they s t i l l c an be done independent ly . T h e first t e r m of the d i f f r ac t i on response for a wavelet W(£)5 for t < a + tw, is 9(1)=ff(l),g(1)=v°pr du (131) T h e 1/s/u — a s i n gu l a r i t y m a y be h a n d l e d i n several d i f ferent ways. T h e crudest w o u l d be to s tar t the c onvo l u t i o n at ft + A t , w h i c h resu l t s i n some a m p l i t u d e loss. A n o t h e r m e t h o d w o u l d invo lve i n teg ra t i n g by par t s , as i n B e r r y h i l l (1979), w h i c h reduces the convo l u t i on of the wavelet a n d the ope ra to r t o the convo l u t i on of the 54 de r i va t i ve of the wavelet w i t h the integra l of the opera to r p lus a co r rec t i on t e r m , as fo l lows. D e n o t i n g the wavelet der i va t i ve by dW(t)/dt — W'(t) 1 W'(t - u) Q(t) = - ^ t a n " 1 t\Ja2 - 0 32 / Ja nR t a n " i\J<*2 - 0\ du. (132) A t h i r d m e t h o d involves us ing the fact t h a t the wavelet is s m o o t h l y v a r y i n g w i t h respect t o the opera to r near the s i ngu l a r i t y so t ha t for the f irst few s a m p l i n g intervals t he wave let may be taken outs ide the in tegra l . If th i s is done j u s t for the first i n te rva l , t h e n , a s s um ing W(t) cons tant on tha t i n te rva l , W K 1 Ja+At (u2 - 02)y/vJ^ a ' + W{t-a) +W(t-a - At) ra + A f du \ 'a [u2 -02)y/u^~cS) r« W(t-u)du Ja+At ( t i 2 - 02)\/u2 - a TIC W{t - a)+W{t - a-At) 2T7R ~ t a n 1 0i V*2 - 01 \ or [a+ At)2 (133) T h e second t e r m above cou l d be reduced f u r t he r by app rox ima t i on s w h e n a ^> At a n d w h e n 0 is not close to ot. T h e first t e r m may be eva lua ted d i rec t l y , by o r d i -n a r y n u m e r i c a l c onvo l u t i on . T h e co r rec t i on t e r m represents the amoun t of a m p l i t u d e deg r ada t i on p r o d u c e d by the c rude m e t h o d of s t a r t i n g the c o n v o l u t i o n at t — a + At. T h i s p rocedu re cou l d be ex tended to several s a m p l i n g interva l s - i ndeed the choice of where t o sp l i t the convo l u t i on i n teg ra l is a rb i t r a r y . 55 Alternatively, if the entire integral may be discretized into a sum of interval in-tegrals then it is preferable to assume W(t) piecewise linear instead of piecewise constant. This technique may be outlined as follows. Begin with the convolution integral o(t) = \Ta*-ft r w^ -u)du U 7TC Ja (U2 - 32)\/u2 ~ CC2 (134) Discretize by letting t — a = KAt, rk = a + (k — l)At and W(t — rk) = Wk. Assume W(t) is piecewise linear so that on the A;th interval, rk < u < rk+1, W(t — u) = cku-{-dk, where W k + l - W k (Tk{Wk+1-Wk}\ . ck = — and dk = Wk - rkck = Wk - ( — J- . (135) At At Then Q(t) \f<*2 - Pi ^  P+' cku + dk 7TC ^ J r k (U2 - 02)y/u^o72 du \A*2 - 31 £ ( Pick + dk\ ( Pick-dk\ \ 2/3, j \ 20! ; K 2xR ]T {3ick+dk)t<m 1 u + 3i a2 - 0tu du \/u2 — a2 + [PiCk ~ dk) tan 1 yja2 - 01 Vu2 - a2 a2 + 0xu yja2 - 0\y/u2 - a2 * ( 0 l C k + dk)(Tll) - + ( 0 l C k - dk)(Tk{2) - r^) E 2TXR ( 2 ) where T,^1, = tan 1 a2 ± Pin Ja2 - Plyjrl - a2 (136) 56 T h e above s u m , w h i c h is bas i ca l l y equ iva lent t o the i n teg ra t i on by par t s us ing the n u m e r i c a l de r i va t i ve of the wavelet, is the most accura te m e t h o d of c onvo l v i n g w i t h the s i ngu l a r i t y a n d was used t o check our results f r o m the s u b t r a c t i o n m e t h o d . However, due t o its c o m p l e x i t y and dev i a t i on f r o m a s t a n d a r d convo l u t i on , i t was not used i n the final m o d e l l i n g p r o g r a m . If the piecewise l inear i n teg ra t i on is p e r f o r m e d for j u s t the first i n te rva l , t hen the fo l l ow ing more accura te f o r m of E q u a t i o n (133) is ob ta i ned . \Ja2 - ft2 rt du W(t - a-At)-W(t - a) yja.' - pt ft 7TC Ja+ At (u2 - Pl)Vu2 - a2 2cAt W(t - a) + (ft - a ) C l I a 2 - f t ( a + Ai ) t a n ' 2 7 r R {yjot2 - P\y/2aAt - At2 2 * R { yja2 - P\j2aAt - At2 where a = \W[t - a - At) - W(t - a)}/At. A f i na l m e t h o d is one we sha l l c a l l the " s u b t r a c t i o n of s i n gu l a r i t y " m e t h o d ( Y e d l i n , pe r sona l c o m m u n i c a t i o n , 1987), w h i c h , due t o i ts elegance, eff iciency a n d m i n i m u m dev i a t i o n f r o m a s t a n d a r d convo l u t i on , is used i n a l l the examp le ca l cu la t i on s . It is 57 der i ved us ing an app l i c a t i on p f l ' H o p i t a l ' s ru le, as fo l lows. Q(t) \jct2 - f t 2 rt W(t-u)du f J a 7TC J  (u2 - 02)\/u2 - a2 \Ja2-02W{t - a) rt •nc L(t) + \A2 - PI 7TC r Ja [ u 2 - 02 / 4>(u)du, J a + cr Ja2 - 02 rt / tb[u) du 7TC Ja where (138) 4>(u) L(t) = W(t-u)-W(t-a) (u2-/32)v/u2-a2 o, ^ t a n - 1 u > a u = a ^(«-«)£a'' itHi V i 2 and , ft>0 ft = 0 . (139) (140) T h i s w i l l be exp l a i ned i n more de ta i l a n d d i sc re t i zed i n Sect ion 4.4. T h e f ina l ou tpu t w i l l be m u l t i p l i e d by R' to no rma l i z e w i t h respect to the ref lect ion a m p l i t u d e , or by R\ to s imu l a t e the geomet r i ca l sp read ing co r rec t i on used in s t a n d a r d process ing, w h i c h assumes the co r rec ted a r r i va l s are ref lect ions and thus a t i f i c i a l l y magni f ies amp l i t udes on the flanks of the h y p e r b o l a . 4.3 Truncation "ghost" effect removal. C o n v o l u t i o n in tegra l t r u n c a t i o n ghosts are we l l k n o w n , a n d have been exam ined i n the con tex t of d i f f r a c t i on t heo r y by B e r r y h i l l (1977, 1979). T h e y ar ise due to the t r u n c a t i o n of the d i f f r a c t i on ope ra to r at t — T p . S ince i t is des i rab le to evaluate the 58 c o n v o l u t i o n out to t = Tc = Tr> + tw, for t > Tp the convo l u t i on in tegra l takes the f o r m \/«2-/5i2 (TD W(t-u)du r i f . . , 7TC Ja (u 2 - Pi)\Ju2 - a 2 *,.) = ^ /' ( W { ' - ^ . (142) T h i s e r ro r i n teg ra l may be eas i ly eva luated by n o t i n g t h a t for Tp su f f i c ient ly large, t he d i f f r a c t i o n opera to r flattens out and m a y be taken t o be a p p r o x i m a t e l y l inear be tween TD a n d t. Se t t i ng S = \D{t) - D(TD)]/(t - TD), th i s impl ie s t h a t D{u) ~ D{TD) + S{u-TD), T D < u < t . (143) E{t) = D{TD) /"' W(t -u)du + S [* (u- TD)W(t - u) du. (144) JTD JTD N e x t , \etv = t-u, W'(v) = £ W(t) dt, and Wn(t) = /0" WJU) dt, and in teg ra te (144) b y pa r t s to o b t a i n E(t) = D(TD) f DW(v)dv + s[ ° (t - v - TD)W [v) dv Jo Jo = D(TD)W\t -TD) + sl{t-v- T^W^v) 1 T° + f~TD W\v) dv = D{TD)W\t-TD) + SWn{t-TD). (145) T h e n u m e r i c a l integra ls of the wavelet are eva lua ted once on ly a n d s to red i n a vector fo r use th roughou t . T h e a p p l i c a t i o n of the above is d i scussed f u r the r i n Sec t i on 4.4. N o t e t h a t the endpo in t of the a s sumed l inear segment c o u l d be set at Tc i n s tead of t. 59 T h i s w o u l d result in a s l ight increase i n c o m p u t a t i o n a l efficiency bu t w o u l d sacrif ice some accu racy for t >Tp. 4.4 Discretized form of the convolution integral. T h e resu l t s of sect ions 4.2 and 4.3 can be c o m b i n e d into a genera l i zed f o r m of the c o n v o l u t i o n i n teg ra l , Q{t) = W[t)*D[t) \A*2 - 81 ' - Pi f 7TC JML rM2 fw{t - u) - H+{a - t + tw)W{t - a) du ( u 2 - 01) Vu2 - a2 + H+{a-t + tw)L{t)+H.(t-TD)E{t) (146) where Mj = Max(a,t - tw), M2 = M'm(t,TD), (1, x>0 ( 1, x > 0 H+ (x) = I and 7J_ [x) = I [ 0, x < 0 { 0 , x < 0 D i s c r e t i z i n g w i t h s a m p l i n g i n te rva l At, the n u m b e r of wavelet s amp le po in t s is Iw ='{tw/At) + 1 a n d the d i f f r ac t i on ope ra to r is t r u n c a t e d at = a + LD and has ID = [LD I At) + 1 sample po int s . T h e o u t p u t w i l l have a length of LC = LD + tw, w i l l be t r u n c a t e d at Tc = T p + tw and w i l l have IC = ID + IW — 1 sample po in t s . Deno te the d i screte t i m e and the d iscrete wavelet by rk = a + (k — l)At and Wk — Wfa — a). R e t u r n i n g to the i n teg ra l in E q u a t i o n (146), denote t b y Tj and u by r t . T h e n W(t - a) = Wj a n d W(t- u) = W,-_ f c+i. A l s o set Kx = M a x ( l , j - + 1) a n d the uppe r l i m i t K2 = M i n ( / r ; , i ) . T h e d i screte opera to r , " s u b t r a c t e d wave le t " 60 and subtraction correction terms are Dx = 0, Dk U = Wj-t+1 -Wj, for j = 1,..., Iu wj-k+i, for j > Iw. Lfa), for j = l,...,Iw; 0, for j > Iw. k > 2 and (147) (148) (149) For the convolution Dk is truncated at k = i p , but terms from ID to IC are also calcu-lated for use in the truncation correction term, which, using trapezoidal integration, is Ei 0, for j = 1,... ,ID; DlDW}_lD+1 + W»lD+1, for j > ID. where the numerical integrals of the wavelet are (150) ^ = 0 , W\ = A * W\l = 0, Wll = At Z + X>, 3=2 j=2 k — 2,. . . , Iw; /c — 2 , . . . , ./^. (151) (152) W i t h the above definitions the convolution integral (146) becomes Qj = Lj + Ej + At K 2 - l y = i , . . . , / e . ( i 5 3 ) See Figure 6 for a schematic representation of the convolution operation and the associated corrections. 61 In the c o m p u t e r p r o g r a m (avai lable u p o n request ) , i n s tead of p i c k i n g a j and s u m m i n g over values of the opera to r a n d wave let indices we use a more elegant nested l oop f o r m u l a t i o n , s im i l a r t o tha t i n C l a e r b o u t (1976), bu t i n c o r p o r a t i n g the cor rec t ion t e rms v i a s imp le compar i son tests. M\ = a L(t) needed M 2 = t a S L I D I N G W A V E L E T M i = t -tw no co r r e c t i on M 2 = t D i f f r a c t i on ope ra to r Mi = t E(t) needed M 2 = TD -D Figure 6: S chemat i c representat ion of c onvo l u t i o n of a wavelet w i t h the d i f f rac t ion ope ra to r , showing when each of the two cor rect ions appl ies . Fo r opera to r lengths shor ter t h a n or equa l to the wavelet l eng th , b o t h co r rec t i on s m a y app l y at once. Fo r p r a c t i c a l a p p l i c a t i o n the above p rocedu re is repeated for the second d i f f racted t e r m ( i n vo l v i ng /32) and b o t h are m u l t i p l i e d by the co r re spond ing s i g num funct ions . T h e n the ref lected t e r m , w h i c h is j u s t a t ime - sh i f t ed ver s ion of the wavelet mod i f i ed by geomet r i ca l sp read ing , is added. It is best to no rma l i z e a l l amp l i t ude s w i t h respect to the peak re f lect ion a m p l i t u d e , a n d , for re f lec t ion se i smology mode l l i n g , to ignore the d i rec t a r r i v a l . F u t u r e extens ions t o th i s work w i l l be o u t l i n e d i n C h a p t e r 6. 62 5 . E X A M P L E S . 5.1 Single trace examples. F i r s t , we examine some single zero-offset traces to de te rm ine the effectiveness of the c o n v o l u t i o n techn ique. F i gu re 7 shows the 32 H z R i c k e r wavelet (7a) used i n a l l the f o l l ow ing ca l cu la t i on s a n d the second t e r m (7b) of the d i f f r ac t i on opera to r , w i t h p l o t t i n g beg inn ing at t = a + At, for d i s tances of —60 metres (do t ted l ine) , —240 metres (sol id l ine) a n d - 4 2 0 metres (dashed l ine) f r o m the edge, where the minus s i gn means in to the ref lect ion shadow zone, to the left in th i s case, as i n later f igures. W h e n the source-receiver pa i r approaches a l o ca t i on above the edge, the d i f f r ac t i on ope ra to r approaches a de l ta f u n c t i o n , wh i l e f u r t he r f r o m the edge more energy resides i n t he opera to r t a i l , i n agreement w i t h results i n B e r r y h i l l (1977). We now examine the error p r o d u c e d by s t a r t i n g the convo lu t i on at a + At i n s tead of a for the above three operator s . F i gu re s 8(a), 8(b) a n d 8(c) dep ic t t he co r rec ted a n d unco r rec ted vers ions of these convo lu t i on s . Nea r t he edge the er ror is we l l over 100 percent (8a) wh i l e fu r ther f r o m the edge (8c) the er ror is on the order of 20 percent . T h i s aga in ind icates t ha t mos t of the d i f f r a c t i on opera to r energy is in the s i n gu l a r i t y reg ion near the edge a n d the t a i l reg ion f u r t h e r away. T h u s some f o r m of co r r e c t i o n such as ou r s u b t r a c t i o n m e t h o d or B e r r y h i l P s (1977) i n teg ra t i on by pa r t s 63 is essential. 64 J I L 0 . 0 2 0 . 0 5 0 . 0 8 t (sec) Figure 7: (a) T h e 80 msec, 32 H z R i c ke r wavelet used in a l l t race ca l cu l a t i on s , (b) the second t e r m of the po i n t source d i f f r ac t i on ope ra to r f o r d i stances of ( l ) —60 m, (2) —240 m a n d (3) —420 m respect ive ly, w i t h the va lue at the s ingu la r po i n t set to zero for p l o t t i n g purposes. 65 Figure 8: (a,b,c) Convolved diffraction responses for the wavelet and operators in Figure 7. The solid lines represent responses derived by neglecting the effect of the singularity in the operator and starting the convolution one sample point over. The dotted lines represent the same done using the subtraction correction method. 66 F i gu re s 9(a) , (b) and (c) dep ic t the t r u n c a t i o n co r rec t i on for the —240 m operator t r u n c a t e d at lengths of 160 msec, 80 msec a n d 40 msec respect ively. T h e uncor rected vers ions ( so l id l ine) e x h i b i t a p seudo -d i f f r ac t i on t r u n c a t i o n effect w h i c h interferes w i t h the d i f f r a c t i o n wave let for shorter ope ra to r lengths. E x a m i n i n g the co r rec ted versions (offset d o t t e d l ine) , for the 80 msec case (9b) the d i f f r ac t i on wavelet is recovered a lmost pe r fec t l y and even the 40 msec case does not result in m u c h a m p l i t u d e degradat ion . T h e relative er ror w i l l be greater for l oca t ions far f r o m the edge and less fo r locat ions near the edge. S ince the amp l i t ude s are largest near the edge an ope ra to r length of ha l f the wavelet l ength shou ld be acceptab le . In a l l the f o l l ow ing examples we used an ope r a t o r the same length as the wave let , i.e. 80 msec or 20 samples . A s shown by offset dashed l ines, th i s is a great imp rovemen t over the m e t h o d used by B e r r y h i l l (1979), w h i c h reduced the t r u n c a t i o n a r t i f a c t by e xp l o i t i n g a sepa ra t i on between the d i f f r a c t i o n wave let and the a r t i f a c t . A l s o , ou r m e t h o d is m u c h s imp le r t h a n techniques desc r ibed by T h o m s o n and C h a p m a n (1987) t o hand le t r u n c a t e d integra l s . 67 Figure 9: (a,b,c) The convolved response for the wavelet in Figure 7a and the —240 m operator truncated at 160 msec, 80 msec and 40 msec respectively, uncorrected (solid line), versions corrected by our method (dotted line) and versions cor-rected by Berryhill's method (dashed line). The corrected versions are verti-cally offset by 0.05 for clarity. 68 5.2 Example sections with Kirchhoff comparison. We sha l l now do a coup le of examp le sect ions by i t e r a t i n g over s ingle traces. F i g u r e 10 is a zero offset sect ion over a ha l f p lane at a dep th of 750 metres i n a m e d i u m w i t h a ve l oc i t y of 1500 m/s. F i g u r e 10(a) dep ic t s the t o t a l f ie ld exact so lu t i on , F i g u r e 10(b) the K i r c h h o f f t o t a l f ie ld s o l u t i on and F i gu re 10(c) the difference of the two. N o t e the a s s y m m e t r y in 10(a) c ompa red t o 10(b). T h e va r i a t i on observed i n the differ-ence of the two can be exp la i ned by e x a m i n i n g the d i f f r ac t i on te rms i n more deta i l . F i g u r e 11(a) shows our exact d i f f r ac t i on ope ra to r a n d 11(b) the K i r c h h o f f one. T h e di f ference is the same as the difference of the t o t a l f ie ld (10c). T h e s y m m e t r y char -ac te r i s t i c s of the d i f f r ac t i on hype rbo lae are more read i l y appa ren t w i t h the ref lected f ie ld r emoved . O u r d i f f r a c t i o n opera to r may be b roken in to two terms dep i c ted in F i g u r e s 11(c) a n d (d). T h e first t e r m has cons tant po l a r i t y a n d re la t i ve l y cons tant low a m p l i t u d e wh i l e the second t e r m exh ib i t s a po l a r i t y reversa l a n d h i gh amp l i t ude s near the edge. T h e second t e r m is very close to the K i r c h h o f f ope ra to r bu t the dif ference, p l o t t ed in F i g u r e 11(e), shows tha t the K i r c h h o f f ope ra to r differs f r o m th i s second t e r m more on the flanks of the hyperbo la . ) These results agree qua l i t a t i v e l y w i t h e x p e r i m e n t a l a n d f requency d o m a i n results presented i n N a r o d a n d Y e d l i n (1986) a n d Y e d l i n et a l . (1987). 69 1.45-F i g u r e 10: (a) exact ref lected and d i f f rac ted f ie ld , (b) K i r c h h o f f equ iva lent and (c) the di f ference of the two (scaled i n a m p l i t u d e by a f ac to r of 6), fo r zero-offset po i n t source inc idence on a ha l f p lane bu r i ed at a dep th of 750 m i n a m e d i u m w i t h a ve loc i t y of 1500 m/s. T race spac ing is 60 m, the a r row denotes the edge l o ca t i on a n d traces are sh i f ted by —40 msec so t h a t the re f lect ion a r r i v a l t i m e co inc ides w i t h the peak va lue and the d i f f r ac t i on a r r i va l t imes l ie on the do t ted h y p e r b o l a . Un less o therwi se s ta ted , the above apppl ies to la ter f igures as we l l . F i g u r e 11: (Nex t page.) (a) the exact d i f f r ac t i on h y p e r b o l a , (b) the K i r c h h o f f one, (c) the f irst t e r m of the exact h y p e r b o l a (scaled by a f ac to r of 6), (d) the second t e r m of the exact h y p e r b o l a , a n d (e) the difference between (d) a n d (b) (scaled by a fac to r of 20), for the m o d e l of F i gu re 10. 70 71 In F i gu re s 12 and 13 the above ca l cu l a t i on s are repeated for a d e p t h of 225 me-tres. D u e to the w ide angle ape r tu re effect the f i rst and second te rms become equiv -a lent i n a m p l i t u d e m u c h more r a p i d l y a n d thus the h y p e r b o l a a s s ymmet r y is ma r ked (12a,13a). A l s o , the difference (I3e) between the second t e r m of the exact d i f f rac t ion o p e r a t o r and the K i r c h h o f f one is m u c h greater. T h i s ( s ymmet r i ca l ) difference p lus the first t e r m is equ iva lent to the t o t a l f ie ld d i f ference. T h e above b r eakdown of the K i r c h h o f f a p p r o x i m a t i o n for d i f f r a c t i on f r o m an edge agrees w i t h results presented by Jebsen a n d M e d w i n (1982) based on the B i o t and To l s toy (1957) wedge so lu t i on . M a n y exp lo ra t i on i s t s m i gh t argue against the ca l cu l a ted a s y m m e t r y by s t a t i ng t h a t t he s u m of the responses of two c o m p l e m e n t a r y ha l f p lanes mus t equal the response of a who le p lane. However the s u m of the responses of two ha l f p lanes w i t h a c o m m o n edge is equa l to the response of a who le p lane p lus t ha t of the region of ove r lap , a l ine scatterer . T h i s is dep i c t ed s chemat i ca l l y in F i g u r e 14. If t he re f lect ion f r o m the left ha l f p lane is Ri, the re f lect ion f r o m the r i ght ha l f p lane is R2 and the d i f f r a c t i on ope ra to r te rms are Di a n d D2, t h e n R e s p ( l ine scatterer ) = Re sp ( L H P ) + Re sp ( R H P ) — Re sp (whole p lane) = {Rx + D i - D2) + [R2 + A + D2) - (Rx + R2) = 2DX 72 Figure 13: (Overleaf) As for Figure 11, but for a half plane at a depth of 225 m. 73 T h u s the response of a l ine scatterer is j u s t tw i ce the first t e r m of the ha l f p lane d i f f r ac t i on operator . T h i s results in a h y p e r b o l a w i t h constant po l a r i t y t h roughou t , as i n F i g u r e 11(c). I n tu i t i ve ly , f r o m the s y m m e t r y of the d i f f rac t ing l ine, th i s constant p o l a r i t y is correct. Rece iver s equ id i s tant f r o m the l ine scatterer , bu t on oppos i te s ides, receive pulses of the same po la r i ty . These pulses may differ i n a m p l i t u d e and shape unless the source is above the l ine. P r e l i m i n a r y work a imed at m o d e l l i n g by i n t eg r a t i on or s u m m a t i o n over l ine scatterers verif ies th i s result. A s o l u t i on der ived u s i ng the B o r n a p p r o x i m a t i o n ( Lumley , pe r sona l c o m m u n i c a t i o n , 1987) shows some s i m i l a r i t y to our s o l u t i o n and also verif ies the cons tant po l a r i t y a long the hype rbo l a . F i g u r e 14 : Schemat i c dep i c t i o n of a the response of a l ine scatterer as the difference of the s u m of the responses of two c o m p l e m e n t a r y ha l f p lanes and the response of a who le p lane. 75 5.3 Oblique incidence examples. We now exam ine the v a r i a t i on of the examp le sect ion as the angle of the seismic l ine w i t h t he edge var ies f r o m the pe rpend i cu l a r . F i g u r e 15 depicts mode l s w i t h s im i la r pa r amete r s to t ha t of F i gu re 10 except t h a t the source-receiver offset is 240 m and the ang le of the seismic l ine w i t h the n o r m a l to the edge is no longer zero, b u t varies f r o m (a) 20° t h r ough (b) 45° to (c) 70°. C o m p a r i s o n s between these and ear l ier mode l s i nd i ca te t h a t as the o b l i q u i t y angle increases t he effective aper tu re decreases, r e su l t i n g in a na t t e r h y p e r b o l a w i t h l i t t l e moveou t a m p l i t u d e v a r i a t i on . T h u s , for se ismic l ines near l y pa ra l l e l to an edge, the d i f f r ac ted a r r i v a l may easi ly be confused w i t h a re f lect ion. 76 Figure 15: 240 m offset oblique-incidence examples for a half plane at a depth of 750 m, for an transect incidence angle of (a) 20°, (b) 45° and (c) 75°, with other parameters as in earlier figures. 77 5.4 Depth variation. In a s i m i l a r ve in , we exam ine va r i a t i on in d i f f r a c t i on ar r i va l s as the d e p t h to and R M S ve l oc i t y above an edge is var ied in accordance w i t h a rea l i s t i c ea r t h mode l . F i g -ure 16 dep ic t s the responses of ha l f planes at depths a n d R M S ve loc i t ies of (a) 5 k m , 2500 m/sec, (b) 15 k m , 5000 m/sec a n d (c) 30 k m , 6000 m/sec. T h e ca l cu l a ted sec-t i ons are zero-offset a n d have a t race sepa ra t i on of 300 m. D u e to an ape r tu re effect, at g reater dep th the d i f f r a c t i on h y p e r b o l a flattens out , becomes more s y m m e t r i c and exh i b i t s less a m p l i t u d e v a r i a t i o n w i t h moveout . T h u s , conce ivab ly , at depths c o m m o n i n deep c r u s t a l s tudies, d i f f ract ions cou ld be m i s t a k e n for ref lect ions. T h i s war ran t s f u r t h u r i nves t i ga t ion w i t h rea l deep c ru s ta l da ta . It m a y also have some imp l i c a t i on s for d i f f r a c t i on process ing schemes, w h i c h m i gh t no t be as effect ive at great dep th . A c -c o r d i n g t o Y e d l i n (persona l c o m m u n i c a t i o n , 1987), a " r a d i a l t r a ce " p roces s ing scheme m i g h t be usefu l in d i s t i ngu i sh i ng between d i f f rac t ions a n d ref lect ions. 78 Figure 16: Three zero offset sections with a trace separation of 300 m, for depths and RMS velocities of (a) 5 km and 2500 m/sec, (b) 15 km and 5000 m/sec, and (c) 30 km and 6000 m/sec. 79 5.5 Trace gather variation W e now compa re var ious me thod s of d i s p l a y i n g recorded seismic d a t a (gathers) a n d the i r effects on d i f f ract ions . F i g u r e 17(a) dep ic t s a c o m m o n offset sect ion w i t h a source receiver offset of 600 m, w i t h other pa ramete r s as in ear l ier mode l s . A s i d e f r o m the onset t i m e there is very l i t t l e d i f ference. F i g u r e 17(b) dep ic t s a c o m m o n source ga ther w i t h the source at +240 m f r o m the edge. F i g u r e 17(c) is a C D P gathe w i t h the d e p t h po i n t at +240 m f r o m the edge. C o m p a r i n g (a-c), the gather t ype w h i c h best separates the d i f f r a c t i on a n d re f lec t i on a r r i va l s is the c o m m o n offset gather, w h i c h shou ld thus be used i n any d i f f r ac t i on process ing techn iques , in agreement w i t h L a n d a et a l . (1987). Fo r dep th or source po in t s closer to the edge the ref lect ion a n d d i f f r a c t i on t r a ve l - t ime curves w i l l coalesce even fu r the r , wh i l e f u r t he r away they separate more. F i g u r e 17(d) shows the effect of C D P s t a c k i n g on the d i f f r ac t i on wave fo rm, u s ing d i f ferent moveout co r rect ions , for d i s tances f r o m the edge of 200 m ([ l ] ) , 400 m ([2]) a n d 600 m ([3]). T h e so l i d l ine is the cen t r a l t race i n each gather , t he do t ted l ine is t h a t s tacked u s ing d i f f r ac t i on moveout c o r r e c t i on , the dashed l ine is t ha t u s ing the moveou t co r rec t i on for a ref lector a s sumed to be at the d e p t h i nd i ca ted by the d i f f r a c t i o n onset t ime , a n d the cha i ndo t t ed l ine is t ha t co r rec ted u s ing the ref lector moveou t . Fo r larger d i s tances f r o m the edge the l a t te r me thod s b reak down ent i re ly, i n d i c a t i n g t h a t i f d i f f r a c t i on process ing is des i red , d i f f r ac t i on moveout mus t be used. T h u s B e r r y h i l l ' s (1977) a r gument t h a t s tacked d a t a represents d i f f rac t ions accura te l y is i n co r rec t , s ince re f lect ion d a t a is a u t o m a t i c a l l y s tacked acco rd i ng to the ref lect ion moveout . 80 Figure 17: C o m p a r i s o n of var ious types of t race gathers a n d s t a ck i ng procedures, for the m o d e l of F i gu re 10. (a) A c o m m o n offset gather w i t h an offset of 600 m. T h e ar row denotes the edge, (b) A c o m m o n source gather , w i t h the source, deno ted by the arrow, 240 m to the r i gh t of the edge, (c) A C D P gather w i t h the m i d p o i n t 240 m to the r ight of the edge, (d) T y p e s of s t a c k i n g procedures , e xp l a i ned in de ta i l in the t ex t , for C D P ga thered d i f f r a c t i on d a t a at d i stances of [1] -200 m , [2] -400 m a n d [3] -600 m f r o m the edge. F i g u r e 18 : Schemat i c dep i c t i on of a the response of a s t r i p as the difference of the s u m of the responses of two ove r l app i ng ha l f p lanes and the response of a who le p lane. 5.6 S t r i p r e s o l u t i o n e x a m p l e s . We now examine , u s ing the m e t h o d of s upe rpo s i t i on , s t r i p mode l s of v a r y i n g w id th s . F i g u r e 18 shows schemat i ca l l y how the s t r i p m o d e l is b roken down i n to the difference of the s u m of two ha l f p lane responses a n d a who le p lane. F i g u r e 19 shows mode l s for s t r ips w i t h w i d t h s of (a) 600 m, (b) 240 m a n d (c) 60 m. F o r the w ide s t r i p the two d i f f ract ions are d i s t i n c t , b u t as the s t r i p w i d t h decreases t he re f lect ion d i sappears and the two d i f f rac t ions beg in to coalesce. If th i s l i m i t i n g process is c on t i nued a l ine source w i l l resu lt . S ince the d o m i n a n t wavelet f requency is 32 H z and the ve l oc i t y is 1500 m/s, F re sne l zone a n d edge i n t e r ac t i on effects shou ld become i m p o r t a n t for w i d th s less t h a n A = 47 m. 82 Figure 19: Sections with parameters as for the model in Figure 10, for strips with widths of (a) 600 m, (b) 240 m and (c) 60 m. The lower arrows denote the strip edge locations. 83 5.7 Fault simulation. F i n a l l y , in F i g u r e 20 we examine mode l s of a d i p p i n g s t r i p (a) us ing the above-m e n t i o n e d s upe rpo s i t i on m o d e l , and of offset ha l f p lanes (b) w i t h the t r u n c a t i o n po in t s at the same po int s as the ends of the d i p p i n g s t r i p . Fo r the d i p p i n g s t r i p note how the d i f f r ac t i on a m p l i t u d e is skewed, peak i n g to the r ight of the h y p e r b o l a apex due to the d i p of the ha l f p lane. Fo r the offset ha l f p lane m o d e l the geometry a n d the survey geomet ry p rec lude the necess ity of m u l t i p l e d i f f rac t ions or ref lected d i f f rac t i on s . N o r m a l i z a t i o n is done w i t h respect to the ear l iest re f lect ion amp l i t ude . A l t h o u g h it is not rea l l y r igorous, i n (c) we added the two responses, s t r i pp i n g ou t the l ine scatterer c omponen t for m a t c h i n g , for la ter c ompa r i s on w i t h a p p r o x i m a t e step or wedge so lut ions . C o r n e r ref lect ions are not i m p o r t a n t due t o the ob l ique angle and zero offset. 84 F i g u r e 20: Sect ions for: (a) A m o d e l for the response of a s t r i p d i p p i n g f r o m a po int at a depth of 600 m to one at a d e p t h of 900 m, at an angle of about 27°. T h e ar rows denote the l a te ra l pos i t i ons of the t e r m i n a t i o n po in t s , (b) A n offset ha l f p lane m o d e l , sh i f ted by 5 traces f r o m (a), w i t h the ha l f p lanes t e r m i n a t i n g at the endpo in t s of the s t r i p i n (a), (c) T h e s u m of (a) and (b). T h e effects of l ine scatterer over lap are removed at the t e r m i n a t i o n po int s . In a l l three mode l s o the r pa ramete r s are as for F i g u r e 10. 85 6 . C O N C L U S I O N S . 6.1 P o i n t summary. [l] A genera l i zed s o l u t i on fo rmat was deve loped, w h i c h broke the t o t a l field into geomet r i ca l opt i c s a n d d i f f racted terms w i t h o u t re so r t ing to a s y m p t o t i c expan -s ion in te rms of d i f f r ac t i on coeff ic ients, and showed how to der ive exact t ime d o m a i n so lut ions f r o m the co r re spond ing f requency d o m a i n i n teg ra l so lut ions. [2] T h i s genera l i zed so lu t i on f o rma t was app l i ed to p lane wave, l ine source and po in t source so lut ions . T h e re su l t i ng t ime d o m a i n so lut ions , w h i c h were shown to agree w i t h prev ious results, cons i s ted of a d i rect t e r m , a ref lected t e r m and two assoc iated d i f f r ac t i on te rms. [3] T h e d i f f r a c t i on terms reach a m a x i m u m va lue of ha l f the a m p l i t u d e of the geomet r i ca l a s soc ia ted opt ics t e r m at the co r r e spond i ng shadow b o u n d a r y and are m o d u l a t e d by a s i gnum f unc t i on across the boundary . T h i s matches the step f unc t i on d i s con t i nu i t y i n the geomet r i ca l op t i c s ( G O ) field so t h a t the t o t a l f ie ld is cont inuous . A s the edge is app roached , each d i f f r ac t i on t e r m approaches a va lue of ha l f the assoc iated G O t e r m so t h a t , regardless of the app roach angle, the t o t a l field approaches the inc ident field. 86 [4] Fo r the seismic re f lect ion case, the d i f f r ac t i on t e r m assoc iated w i t h the d i rect shadow b o u n d a r y is of r e l a t i ve l y low a m p l i t u d e , but is i m p o r t a n t because, due to its cons tant po l a r i t y , i t enforces a h y p e r b o l a a s s ymmet ry w h e n added t o the o ther t e r m , w h i c h is a m p l i t u d e s y m m e t r i c and has a po l a r i t y reversa l at the apex. A m p l i t u d e s are p r ed i c t ed to be h igher on the ref lector s ide of the edge. [5] T h e low a m p l i t u d e t e r m is equ iva lent t o ha l f the response of a l ine scatterer , and w o u l d be m u c h s t ronger near the d i rec t shadow boundary , w h i c h m i gh t be observed i n V S P or d i f f r a c t i on tomography . [6] K i r c h h o f f so lut ions p red i c t the second d i f f r ac t i on t e r m , a l t hough w i t h some phase d i s t o r t i on , and w i t h low a m p l i t u d e on the f lanks. S ince the f irst t e r m is o m i t t e d , K i r c h h o f f theory i n co r rec t l y p red i c t s a perfect 180° phase r o t a t i o n at the edge and an a m p l i t u d e s y m m e t r i c h ype rbo l a . T h i s was e x a m i n e d i n the examp le s , wh i ch showed t ha t the differences between the K i r c h h o f f and exact so lut ions become greater at sha l lower depths . [7] T w o new n u m e r i c a l a p p l i c a t i o n techn iques have been deve loped for convo lu -t i o n w i t h a l/yjt ~ to s i n gu l a r i t y and for r emov i n g integra l t r u n c a t i o n ghosts. These demons t ra te some imp rovemen t over techniques presented by B e r r y h i l l (1977). [8] D u e to an ape r tu re effect, sect ion examp le s show a decrease in a s s ymmet ry , flattening of the h y p e r b o l a a n d decrease in the rate of a m p l i t u d e decay w i t h moveout as e i ther d e p t h or the angle of the se i smic l ine w i t h the n o r m a l to the edge increases. T h u s , for g reat depths or for l ines near l y p a r a l l e l to the edge, d i f f rac t ions m i gh t be m i s t a k e n l y i n te rp re ted as ref lect ions. 87 [9] C o m m o n depth po in t ( C D P ) s tack ing d i s tor t s d i f f ract ions on the flanks of the h y p e r b o l a , as i t is i n tended to do. Fo r ana lys i s of d i f f rac t ions , a d i f f rac t ion s t a ck i ng p rocedure is best. If a s imu l a ted zero offset sect ion is des i red, C D P s t a ck i n g us ing d i f f rac t ion moveout shou ld be pe r f o rmed . O the rw i se , c o m m o n offset ( C O ) gathers shou ld be used. [10] S t r i p mode l s show d i s t i nc t a r r i va l s for w i d e s t r ip s , b u t for na r r ow st r ips the d i f f rac t i on s beg in to interfere a n d app roach the response of a l ine source. [ l l ] D i p p i n g edges show a peak d i f f r ac t i on a m p l i t u d e a n d p o l a r i t y rever sa l wh i ch is skewed by the same amoun t as the edge n o r m a l a n d no longer fa l l s at (ap-p r o x i m a t e l y ) the apex of the hype rbo l a . 6.2 Discussion. T h e m a i n i m p l i c a t i o n s for se ismic i n t e rp r e t a t i on were t ha t a s s y m m e t r y effects show up at aper tu res of 20°, a n d tha t , for graet depths or for se i smic l ines at s m a l l angles to the edge d i f f ract ions may easi ly be in te rp re ted as ref lect ions. T h e exact so lut ions and p r a c t i c a l a p p l i c a t i o n techn iques presented in th i s thesis are in a f o r m easi ly c om-pa red to rea l d a t a , and may resu l t in improvement s in d i f f r a c t i on i n t e rp r e t a t i on and proces s ing . A condensed vers ion of th i s thesis has been s u b m i t t e d to Geophys. J. R. astr. Soc. ( Da l t on and Y e d l i n , 1988). 88 6.3 Areas for future research. [l] W o r k is p roceed ing on f in i te impu l se response ( F I R ) and recur s i ve in f in i te impu l se response ( I IR or A R M A ) filter o p t i m i z a t i o n , l inear p r o g r a m m i n g and c on t i nua t i on techn iques to reduce c o m p u t a t i o n a l expense even f u r t h u r and to f a c i l i t a t e eva lua t i on of responses for more c o m p l i c a t e d model s , pos s ib l y even 3-D ones. T h i s c o u l d cu t convo l u t i on costs even fu r ther . P r e l i m i n a r y wo r k looks encou rag ing ( D a l t o n 1987). [2] T h e p r a c t i c a l a p p l i c a t i o n techniques deve loped in C h a p t e r 4 c a n be app l i ed t o genera l wedge so lu t ions , such as those by Felsen and M a r c u v i t z (1973) and H u t t o n (1987). U s i n g an image p rocedure deve loped by Z h a n g a n d J u l l (per-sona l c o m m u n i c a t i o n , 1987), th i s cou ld be ex tended t o step so lu t ions . [3] U s i n g the genera l i zed approach i n t r oduced in sect ion 2.1, research is be ing c o n d u c t e d in to so lu t ions for wedges a n d steps. A l s o , i n teg ra t i on over l ine scat-terers or s upe rpo s i t i o n of s t r ips may be used for c omp lex mode l s . T h i s is not r i gorous b u t it m i g h t wo r k , l ike m u c h of geophys ics. [4] E x a m i n a t i o n of the effect of m u l t i p l e d i f f rac t ions shou ld be c a r r i e d ou t . P r e -l i m i n a r y work shows t ha t they w i l l be i m p o r t a n t on ly w h e n one d i f f r ac to r fal ls near the shadow b o u n d a r y of another one. D u e to genera l ly low re f lec t i on co-eff ic ients, m u l t i p l e d i f f rac t ions w i l l p r obab l y be u n i m p o r t a n t i n se ismology, bu t they w i l l be i m p o r t a n t i n e lec t romagnet i c work . [5] G a u s s i a n b e a m mode l s (Cerveny 1985) m a y be eva luated by ana l y t i c con-t i n u a t i o n of the exact d i f f r ac t i on so lut ions t o c o m p l e x source pos i t i on s , as i n 89 Felsen (1984) or W u (1985). T h i s w i l l he lp in s imu l a t i on of rea l sources. [6] A check s hou ld be made to see w h e n e l a s t odynam ic so lut ions are necessary a n d to des ign cor rect ions for any P - S V coup l i n g . [7] T h e E M d ipo le s o l u t i on shou ld be deve loped f u r the r a n d the acoust ic ana -logue der i ved. E l e c t r omagne t i c so lu t ions for the po in t source and the d ipo le cou ld be us ing i n geophys ica l p r o spec t i n g studies or i n studies of s ca t te r i ng f r o m the s u b d u c t i n g l i thosphere. T h e acoust ic d ipo le so lu t i on w i l l he lp i n s imu l a t i n g rea l , d i r e c t i o n a l ma r i ne sources. [8] S pa t i a l i n teg ra t i on over po i n t source so lut ions cou ld also be used to s imu -late rea l sources. Fo r a cu r ved or segmented f au l t i nvo l ved in a n ea r thquake, s ca t te r i ng of ea r thquake energy may be i m p o r t a n t . [9] A more rea l i s t i c wavelet s hou ld be used ins tead of the R i c k e r one for deta i led m o d e l l i n g of rea l da ta . A l s o , the phase characte r i s t i c s of the d i f f r ac t i on opera to r shou ld be e x a m i n e d . [10] Improvement s c ou l d be m a d e on P h a d k e and K a n a s e w i c h ' s (1987) c o m -m o n f au l t s t a ck i ng m e t h o d and on L a n d a et al. 's (1987) d i f f r a c t i on cor re la t ion m e t h o d , by i n c o r p o r a t i n g the exact phase v a r i a t i o n in to the stack. [ l l ] M i g r a t i o n rout ines m a y be deve loped by extens ion of K i r c h h o f f m i g r a t i o n and d ip moveou t and d i f f r ac t i on stack procedures . 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