@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Earth, Ocean and Atmospheric Sciences, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Cabrera, Jose Julian"@en ; dcterms:issued "2010-05-10T03:02:09Z"@en, "1983"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The plane-wave decomposition of the vertical displacement component of the spherical-wave field corresponding to a compressional point source is solved as a set of inverse problems. The solution utilizes the power and stability of the Backus & Gilbert (smallest and flattest) model-construction techniques, and achieves computational efficiency through the use of analytical solutions of the integrals which are involved. The theory and algorithms developed in this work allow stable and efficient reconstruction of the spherical-wave field from a relatively sparse set of their plane-wave components. However, the algorithms do not formally conserve the correct amplitudes of the seismograms. Comparison of the algorithms with direct integration of the Hankel transform shows very little or no advantage for the transformation from the time-distance (t-x) domain to the delay time - angle of emergence (τ-γ) domain if the seismograms are equi-sampled spatially. However, for cases where the observed seismograms are not equally spaced and for the transformation τ-γto t-x, the proposed schemes are superior to the direct integration of the Hankel transform. Applicability of the algorithms to reflection seismology is demonstrated via the solution to the problem of trace interpolation and that of separation of converted S modes from other modes presented in common-source gathers. In both cases the application of the algorithms to a set of synthetic reflection seismograms yields satisfactory results."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/24556?expand=metadata"@en ; skos:note "PLANE-WAVE DECOMPOSITION AND RECONSTRUCTION OF SPHERICAL-WAVE SEISMOGRAMS AS A LINEAR INVERSE PROBLEM by JOSE JULIAN CABRERA B.Sc. E n g i n e e r i n g G e o p h y s i c s , I n s t i t u t e * P o l i t e c n i c o N a c i o n a l (Mexico),1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department Of G e o p h y s i c s And Astronomy) We accept t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December 1983 © Jose J u l i a n C a b r e r a , 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C o lumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of G e o p h y s i c s And Astronomy The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: 5 December 1983 ABSTRACT t The plane-wave d e c o m p o s i t i o n of the v e r t i c a l d i s p l a c e m e n t component of the s p h e r i c a l - w a v e f i e l d c o r r e s p o n d i n g t o a c o m p r e s s i o n a l p o i n t source i s s o l v e d as a s e t of i n v e r s e problems. The s o l u t i o n u t i l i z e s the power and s t a b i l i t y of the Backus & G i l b e r t ( s m a l l e s t and f l a t t e s t ) m o d e l - c o n s t r u c t i o n t e c h n i q u e s , and a c h i e v e s c o m p u t a t i o n a l e f f i c i e n c y t h r ough the use of a n a l y t i c a l s o l u t i o n s of the i n t e g r a l s which a r e i n v o l v e d . The t h e o r y and a l g o r i t h m s d e v e l o p e d i n t h i s work a l l o w s t a b l e and e f f i c i e n t r e c o n s t r u c t i o n of the s p h e r i c a l - w a v e f i e l d from a r e l a t i v e l y s p a r s e s e t of t h e i r plane-wave components. However, the a l g o r i t h m s do not f o r m a l l y c o n s e r v e the c o r r e c t a m p l i t u d e s of the seismograms. Comparison of the a l g o r i t h m s w i t h d i r e c t i n t e g r a t i o n of the Hankel t r a n s f o r m shows v e r y l i t t l e or no advantage f o r the t r a n s f o r m a t i o n from the t i m e - d i s t a n c e ( t - x ) domain t o the d e l a y time - a n g l e of emergence (r-y) domain i f the seismograms a r e eq u i - s a m p l e d s p a t i a l l y . However, f o r cases where the observed seismograms a r e not e q u a l l y spaced and f o r the t r a n s f o r m a t i o n r-y t o t - x , the proposed schemes a r e s u p e r i o r to the d i r e c t i n t e g r a t i o n of the Hankel t r a n s f o r m . A p p l i c a b i l i t y of the a l g o r i t h m s t o r e f l e c t i o n s e i s m o l o g y i s demonstrated v i a the s o l u t i o n t o the problem of t r a c e i n t e r p o l a t i o n and t h a t of s e p a r a t i o n of c o n v e r t e d S modes from o t h e r modes p r e s e n t e d i n common-source g a t h e r s . In both cases the a p p l i c a t i o n of the a l g o r i t h m s t o a s e t of s y n t h e t i c r e f l e c t i o n seismograms y i e l d s s a t i s f a c t o r y r e s u l t s . i v TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS X 1 INTRODUCTION TO THE PLANE-WAVE DECOMPOSITION PROBLEM 1.1 E x p a n s i o n of a Sp h e r i c a l - W a v e F i e l d i n Terms of P l a n e and C y l i n d r i c a l Waves: B a s i c Development 1 1.2 On Plane-Wave D e c o m p o s i t i o n of D i g i t a l Data 15 2 BACKUS AND GILBERT FORMULATION OF THE PROBLEM 2.1 Forward T r a n s f o r m : C o n s t r u c t i o n of the Plane-Wave Components of Sph e r i c a l - W a v e Seismograms 20 2.1.1 S m a l l e s t model c o n s t r u c t i o n 21 2.1.2 F l a t t e s t model c o n s t r u c t i o n 25 V 2.2 I n v e r s e T r a n s f o r m : R e c o n s t r u c t i o n of the S p h e r i c a l -Wave F i e l d from i t s Plane-Wave Components 28 2.2.1 S m a l l e s t model c o n s t r u c t i o n 30 2.2.2 F l a t t e s t model c o n s t r u c t i o n 30 3 EXAMPLES 3.1 I n t r o d u c t o r y Comments 32 3.2 S e p a r a t i o n of C o n v e r t e d S Modes 34 3.3 White Nois e and C o n s t r u c t i o n Q u a l i t y 52 3.4 Angul a r Sampling and Trace I n t e r p o l a t i o n 56 3.4.1 S h a l l o w zone of i n t e r e s t 57 3.4.2 Trace i n t e r p o l a t i o n 61 4 COMPUTATIONAL CONSIDERATIONS 66 5 SUMMARY 69 BIBLIOGRAPHY 71 APPENDIX A X 2 V a l u e and O b s e r v a t i o n a l E r r o r s i n Model C o n s t r u c t i o n 73 APPENDIX B E f f e c t s of the W e i g h t i n g F u n c t i o n Q and S t a n d a r d D e v i a t i o n V a l u e s on Model C o n s t r u c t i o n 76 v i APPENDIX C Inner Product Matrix f o r the Forward Smallest Model C o n s t r u c t i o n 86 APPENDIX D The Forward F l a t t e s t Model: G l o b a l Development 87 APPENDIX E Inner Product M a t r i c e s for the Inverse Smallest and F l a t t e s t Model C o n s t r u c t i o n s 91 v i i LIST OF TABLES I . B a s i c Steps f o r PWD of a Common-Source Gather 17 I i . P r o c e s s i n g Times f o r the PWD A l g o r i t h m s 66 v i i i LIST OF FIGURES 1.1a P r o p a g a t i n g s p h e r i c a l - w a v e f r o n t 4 1.1b Plane waves w i t h d i f f e r e n t p r o p a g a t i o n v e l o c i t i e s ... 5 1.2 Mode e x p a n s i o n i n terms of p l a n e waves 6 1.3 D e f i n i t i o n of the wavenumber components 7 1.4 D i s p l a c e m e n t p o t e n t i a l as a s u p e r p o s i t i o n of p l a n e waves 9 3.1 Model used f o r g e n e r a t i n g the s y n t h e t i c seimograms .. 35 3.2a P-P seismograms 37 3.2b P-S seismograms 38 3.3 PP+PS seismograms 39 3.4 S i g n a l s w i t h l a r g e and s m a l l moveouts 40 3.5 P r e d i c t e d t i m e s f o r PWD components 42 3.6 PWS Hankel t r a n s f o r m 43 3.7 PWS c o n v e r t e d S zone 44 3.8 C o n v e r t e d S modes. Hankel-Hankel 46 3.9 C o n v e r t e d S modes. S m a l l e s t - S m a l l e s t 47 3.10 C o n v e r t e d S modes. F l a t t e s t - F l a t t e s t 48 3.11 C o n v e r t e d S zone. H a n k e l - F l a t t e s t 50 3.12 C o n v e r t e d S zone. S m a l l e s t - S m a l l e s t 51 3.13 PP+PS seismograms. A d d i t i v e random n o i s e 53 3.14 C o n v e r t e d S modes. S m a l l e s t - S m a l l e s t . ( N o i s y seismograms) 54 i x 3.15 Co n v e r t e d S modes. F l a t t e s t - F l a t t e s t ( N o i s y seismograms) 55 3.16 PP+PS seismograms. ( N e a r - o f f s e t r e s a m p l i n g ) 58 3.17 PWS s m a l l e s t model. ( N e a r - o f f s e t r e s a m p l i n g ) 59 3.18 PWS f l a t t e s t model. ( N e a r - o f f s e t r e s a m p l i n g ) 60 3.19 PP+PS seismograms. Uneven Sp a c i n g 63 3.20 I n t e r p o l a t e d S modes. S m a l l e s t - S m a l l e s t 64 3.21 I n t e r p o l a t e d S modes. F l a t t e s t - F l a t t e s t 65 B.1 M o d i f i e d B e s s e l f u n c t i o n s K 0 and K, 77 B.2 The c o n d i t i o n number of the i n n e r p r o d u c t m a t r i x .... 80 B.3 E i g e n v e c t o r s f o r the ( s m a l l e s t ) i n n e r p r o d u c t m a t r i x 81 B.4 The 25th b a s i s f u n c t i o n 82 B.5 E f f e c t of s t a n d a r d d e v i a t i o n s on the b a s i s f u n c t i o n s 85 X ACKNOWLEDGEMENTS I am d e e p l y g r a t e f u l t o my ] /] 1 ~j ^ P (dear, f r i e n d ) Shlomo Levy f o r a l l h i s v a l u a b l e a d v i c e , knowledge and moral support d u r i n g the e n t i r e development of t h i s t h e s i s . I r e g a r d him w i t h p r o f o u n d a f f e c t i o n and look forward t o a p e r d u r a b l e f r i e n d s h i p . I t i s my p l e a s u r e t o warmly acknowledge Dr. Ron Clowes, my s u p e r v i s o r , f o r h i s wise d i r e c t i o n and c o n t i n u o u s encouragement. C e r t a i n l y I have deve l o p e d much r e s p e c t f o r Ron. I t i s a b i t d i f f i c u l t t o e x p r e s s , i n a language o t h e r than my n a t i v e , my g r a t i t u d e t o Dr. Doug Oldenburg. He has p r o v i d e d me w i t h guidance and c o u n s e l l i n g , and shared h i s superb knowledge of I n v e r s e Theory. I hope t o c o n t i n u e working w i t h Doug both as st u d e n t and as \"amigo\". I w i s h t o acknowledge the generous a d v i c e of Dr. George Bluman of the Department of Mathematics a t UBC, f o r h i s v e r y h e l p f u l (and ca l m i n g ) d i s c u s s i o n s , p a r t i c u l a r l y i n r e l a t i o n t o i n t e g r a l s of a r b i t r a r y and m o d i f i e d B e s s e l f u n c t i o n s . E n t h u s i a s t i c thanks a re a l s o due t o M i c h a e l S h l a x , K e r r y S t i n s o n , Kenneth W h i t t a l l and Ian Jones f o r p r o v i d i n g me w i t h so much i n s i g h t f u l a d v i c e . x i I w ish I c o u l d c o n t i n u e l i s t i n g a l l the e x t r a o r d i n a r y p e o p l e who have c o n t r i b u t e d t o the n i c e environment i n which I have s t u d i e d , but space p r e c l u d e s me from d o i n g so. However, I would p a r t i c u l a r l y l i k e t o mention Lynda F i s k , Mark Lane, Don White and Gemma Jones. My M a s t e r ' s s t u d i e s were s u p p o r t e d by a graduate s c h o l a r s h i p from the S c i e n c e and Technology N a t i o n a l C o u n c i l (CONACYT) of Mexico, and O p e r a t i n g Grant A7707 from the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l of Canada. LA MEMORI A DE MEMITO 1 1. INTRODUCTION TO THE PLANE-WAVE DECOMPOSITION PROBLEM 1.1 E x p a n s i o n of a Spherical-Wave F i e l d i n Terms of P l a n e and C y l i n d r i c a l Waves: B a s i c Development For many y e a r s the study of wave f i e l d s produced by s p h e r i c a l waves t r a v e l l i n g i n s i m p l e e a r t h models has been f a c i l i t a t e d by a n a l y s i s of the p l a n e waves r e p r e s e n t i n g the o r i g i n a l s p h e r i c a l waves. Indeed, f o r a medium c o n s i s t i n g of homogeneous l a y e r s , r e f l e c t i o n , r e f r a c t i o n and mode c o n v e r s i o n a t l a y e r i n t e r f a c e s i s s i m p l e r t o i n v e s t i g a t e u s i n g p l a n e r a t h e r than s p h e r i c a l waves. For example, the method of computing s y n t h e t i c seismograms due t o Fuchs and M u l l e r (1971) i s based on s o l v i n g the wave-propagation problem f o r p l a n e waves and then s u p e r i m p o s i n g the plane-wave s o l u t i o n s t o o b t a i n the s p h e r i c a l -wave f i e l d ( i . e . the s y n t h e t i c seismograms). The r e p r e s e n t a t i o n of a s c a l a r , t i me-harmonic, s p h e r i c a l wave f i e l d (SWF) as a s u p e r p o s i t i o n of p l a n e waves i s w e l l documented i n the s e i s m i c and e l e c t r o m a g n e t i c l i t e r a t u r e ; see f o r example, S t r a t t o n (1941), Bath (1968), Goodman (1968), Born and Wolf (1980), and A k i and R i c h a r d s (1980). L u c i d t r e a t m e n t s of the g e n e r a l t h e o r y a r e g i v e n by B r e k h o v s k i k h (1960), and Devaney and Sherman (1973), and some i n s i g h t f u l a p p l i c a t i o n s a r e 2 p r e s e n t e d by Asby and Wolf (1971), M u l l e r (1971), and T r e i t e l e t a l . (1982). F o l l o w i n g A k i and R i c h a r d s (1980; ch.6) we i n t r o d u c e the Weyl plane-wave and the Sommerfeld c y l i n d r i c a l - w a v e e x p a n s i o n s of a SWF. An o u t l i n e of the m a t h e m a t i c a l s t e p s n e c e s s a r y f o r o b t a i n i n g such r e p r e s e n t a t i o n s f o l l o w s . C o n s i d e r the problem of a p o i n t source at x=0 r a d i a t i n g c o m p r e s s i o n a l waves i n a homogeneous, i s o t r o p i c and unbounded medium. Given t h a t the s o u r c e e x h i b i t s time dependence of the form e x p [ - i c j t ] (u> i s an a r b i t r a r y a n g u l a r f r e q u e n c y ) , compressional-wave p r o p a g a t i o n may be d e s c r i b e d by the ( s c a l a r ) d i s p l a c e m e n t p o t e n t i a l /d2t - V 2 V 2 0 = 4T T V p 2 6 ( x ) e x p t - i w t ] (1-1) where V P r e p r e s e n t s the P-wave v e l o c i t y of the medium. The space-time s o l u t i o n t o (1-1) i s * ( x , t ) = [ 1 / R ] e x p [ i o ; ( R / V P - t ) ] (1-2) w i t h , A A A x = x i + y j + z k R = / x 2 +y 2 + z 2 E q u a t i o n (1-1) may a l s o be s o l v e d u s i n g F o u r i e r t r a n s f o r m methods, i n which case the wavenumber-time s o l u t i o n reads as tf(k,t) = [4irVp 2 / ( k 2 V p 2-£j 2) ] e x p [ - i u t ] (1-3) 3 w i t h k = kx i + k 3 j + k , The r e l a t i o n between (x,t) and (x,t) = [1/R]exp[ i u ( R / V p - t ) ] = [ 1/2TT 2 ] e x p [ - i u t ] / / / [ l / ( k 2 - o ; 2 / V P 2 ) ]exp[ i k - x]dk •too (1-4) -co Equat i o n (1-4) r e p r e s e n t s o s p h e r i c a l wave (x,t) t r a v e l l i n g w i t h a c o n s t a n t v e l o c i t y v p as a s u p e r p o s i t i o n of an i n f i n i t e number of homogeneous p l a n e waves, each of which p r o p a g a t e s w i t h a v e l o c i t y w/k (see F i g u r e s 1.1a and 1.1b). Because 00 and n e g a t i v e f o r z<0 ( n o t i c e t h a t we have e l i m i n a t e d the terms e x p { - i w t ] and e x p l i c i t l y s t a t e d the CJ dependence i n the argument of , each p l a n e wave has an a r b i t r a r y v e l o c i t y V;''=a>/k; , where k;=|kf |. Thus, f o r kbG. So f a r we have d e a l t w i t h an i m p u l s i v e p o i n t source a t x=0. For c ases i n which the p o i n t source has a time dependence F ( t ) w i t h a spectrum F ( C J ) , e q u a t i o n ( 1 - 6 ) i s m o d i f i e d t o 0(o),r,z) = JF(CJ) [ 1 / i k j ]exp[ ik„ z ] k r J 0 ( k r r )dk r ( 1 - 8 ) o Up t o t h i s p o i n t we have c o n s i d e r e d an unbounded homogeneous space. For a medium c o n s i s t i n g of a sequence of homogeneous l a y e r s , the t o t a l d i s p l a c e m e n t p o t e n t i a l a t an o b s e r v a t i o n p o i n t P l o c a t e d a t some depth z i s g i v e n by the 8 c o n t r i b u t i o n from the d i r e c t and r e f l e c t e d p o t e n t i a l s ( r e c a l l t h a t t h e s e p o t e n t i a l s propagate as s p h e r i c a l wave f r o n t s ; see F i g u r e 1.4). In t h i s c a s e , t h e n , the c o r r e s p o n d i n g plane waves ( i n view of 1-5) or c y l i n d r i c a l waves ( i n view of 1-6) a r e weigh t e d by plane-wave r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s . A d d i t i o n a l l y , w i t h i n the i t t l a y e r each p l a n e (or c y l i n d r i c a l ) wave i s c h a r a c t e r i z e d by the v e r t i c a l wavenumber k,r>} = y/cj2/V2 ~ k 2 = ucos (7 ; p )/Vrp (1 - 9a) f o r P waves. By s i m i l a r arguments the wavenumber f o r a S wave, k*; = V/CJVV;5 2 - kr2 = ucos(7; s )/V:S (1-9b) may be found. The h o r i z o n t a l wavenumber i s k r= ws i n (y* )/V p, and a s u p e r s c r i p t l e t t e r denotes the wave mode. The d i s p l a c e m e n t p o t e n t i a l a t P ( r , z ) i s then o b t a i n e d from (see F i g u r e 1.4)

° + 01+tf>2 (1-10) where each p o t e n t i a l i s e x p r e s s e d as ( c f . e q u a t i o n 1-8) 0°(cj,r,z)= /F(o)) [ 1/ik; ] e x p [ i k j ; 1 z ] k r J 0 ( k r r ) d k r (1 -11 a) o 0 1 ( c j , r , z ) = /F(o))A[ 1/ik; ] e x p [ i k p l / 1 h 1 + i k P i l (h,-z) ] o k r J 0 ( k r r ) d k r (1-1 l b ) 9 vfv,' Source, vfv* Plr,z) z = 0 O r V,P 7 < vP (C) fe) FIGURE 1.4 Displacement p o t e n t i a l at the ob s e r v a t i o n p o i n t P ( r , z ) as a s u p e r p o s i t i o n of plane waves, (a) The displacement p o t e n t i a l at P i s given by 1 +°, i n (d) the plane wave a f f e c t e d by the r e f l e c t i o n c o e f f i c i e n t A c o n t r i b u t e s to the p o t e n t i a l 0 1, and in (e) the plane wave a f f e c t e d by the t r a n s m i s s i o n c o e f f i c i e n t s B and D and the r e f l e c t i o n c o e f f i c i e n t C c o n t r i b u t e s to 2 ( u , r , z) = fF(u)BCDexp[ i k p y,h , + i k p 2 2 h 2 + i k£ , (h ,-z ) ] [ i / i k * a ] k r J 0 ( k r r ) d k r . (1-1 1c> The v e r t i c a l wavenumbers k^, and k p ; 2 a r e g i v e n by (1-9a) and, because the c o m p r e s s i o n a l source i s l o c a t e d i n the f i r s t -l a y e r , k i = k i ( 1 . A and C a r e plane-wave r e f l e c t i o n c o e f f i c i e n t s , w h i l e B and D are plane-wave t r a n s m i s s i o n c o e f f i c i e n t s . They depend on the e l a s t i c p r o p e r t i e s of the medium and on the ray parameter p ( r e c a l l t h a t p=cjkr =cjsin ( 7 , )/V P ) . From (1-10) and (1-11) the f u n c t i o n d e f i n e d by V ( c j , k r ; z ) = F(co)exp[ i k i , z ] + F (o>) Aexp[ i k\\t, h , + i k^ , (h ,-z ) ] + F(w)BCDexp[ i k ; - 1 h , + i k p , 2 2 h 2 + i k P . 1 (h,-z) ] (1-12) w i l l be u n d e r s t o o d as the spectrum of the p l a n e waves d e f i n e d by k r =cjsin {y* )/V p . For a g i v e n a n g u l a r f r e q u e n c y w, V(w,k r;z) g i v e s both the plane-wave c o n t r i b u t i o n from the b o u n d a r i e s between l a y e r s , and the v e r t i c a l phase d e l a y t h a t the (homogeneous) p l a n e waves a c q u i r e d i n each l a y e r . I f k\\x i s i m a g i n a r y , an inhomogeneous p l a n e wave w i l l p r opagate h o r i z o n t a l l y i n the i 1 * l a y e r , and the c o n t r i b u t i o n from the bottom i n t e r f a c e t o t h i s l a y e r w i l l have an e x p o n e n t i a l a t t e n u a t i n g term. T h e r e f o r e , w i t h r e f e r e n c e t o F i g u r e 1.4, e q u a t i o n (1-12) g i v e s the response from a d i r e c t p l a n e wave ( f i r s t term . i n the l e f t hand s i d e ) , from a p l a n e wave r e f l e c t e d from the f i r s t 11 boundary (second term) and from a p l a n e wave r e f l e c t e d from the second boundary ( t h i r d t e r m ) . N o t i c e t h a t these p l a n e waves form p a r t of a system of p l a n e waves s h a r i n g the same h o r i z o n t a l wavenumber. In t h i s c o n t e x t , t h e n , V(a>,k r;z) i s viewed as the spectrum of a plane-wave seismogram e q u i v a l e n t l y d e f i n e d by k r, P or 7f, . Some a d d i t i o n a l i n s i g h t i n t o the plane-wave n a t u r e of (1-12) may be seen i n the time domain. B e f o r e i n v e r s e - F o u r i e r t r a n s f o r m i n g t h i s e q u a t i o n , i t i s c o n v e n i e n t t o make the s u b s t i t u t i o n (see 1-9a) k\\- = wcos(7, )/V;p i = 1 ,2 Then, we may r e w r i t e (1-12) as V(u,y\\;z) = F(u>)exp[ icjzcos ( 7 p , )/V p ] + F(u.)Aexp[iw{h,cos (7 i )/V p + ( h , - z ) c o s ( 7 i )/v'} ] + F(w)BCDexp[ i c j { h ! C O s ( 7 ' ) / V p + 2 h 2 c o s ( 7 f 2 )/Vp2 + ( h 1 - z ) c o s ( 7 e i ) / V ? 1 } ] . (1-13) N o t i c e t h a t we have now used y\\ t o c h a r a c t e r i z e V ( & j , k r ; z ) . For homogeneous p l a n e waves ( i . e . f o r 7.p r e a l ) , i n v e r s e F o u r i e r t r a n s f o r m of (1-13) g i v e s 1 2 V C t ^ w - z ) = F ( t - z c o s ( 7 P i )/V\\) + A F ( t - { h t C O s O ' J / V j + d ^ - z J c o s f T p / V ' } ) + BCD F ( t - {h,cos(7 f, )/V , , 1+2h 2cos(7 ? 2)/V2 + ( h , - z ) c o s ( 7 ? 1 ) / V , } ] . (1-14) For a g i v e n e a r t h model, t h i s e q u a t i o n demonstrates t h a t the p l a n e wave seismogram V( t , 7 ? , ; z ) i s g i v e n by the source f u n c t i o n F ( t ) a p p e a r i n g a t d e l a y times dependent on the c o s i n e s of the a n g l e s of p r o p a g a t i o n of p l a n e waves w i t h i n each l a y e r . F u r t h e r , t h i s source f u n c t i o n i s weighted by plane-wave r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s . Thus, the (homogeneous) plane-wave response f o r the d i r e c t p o t e n t i a l <£°(t,r;z) c o r r e s p o n d i n g t o the a n g l e i s S i m i l a r l y , the plane-wave response f o r the d i s p l a c e m e n t 0 ' ( t , r ; z ) i s g i v e n by N o t i c e t h a t we have used \"plane-wave response\" t o d e s c r i b e those time f u n c t i o n s whose s p e c t r a c o n s t i t u t e complex w e i g h t s f o r p l a n e or c y l i n d r i c a l waves (see e q u a t i o n s 1-11). I f the o b s e r v a t i o n p o i n t P ( r , z ) i s a t the s u r f a c e z=0, the plane-wave response f o r the d i r e c t p o t e n t i a l i s s i m p l y the g 0(t,7 r,.;z) = F ( t - t d ) = F ( t - z c o s ( V i ) / V ^ ) (1-15a) g M t , 7 i ; z ) = A F ( t - t A ) = A F ( t - {h,cos(Vi )/Vi + ( h , - z ) c o s ( 7 F , )/V ?,}). (1-15b) 1 3 s o u r c e f u n c t i o n w i t h no d e l a y t i m e , i . e . g°(t,7^;z) = F ( t ) f o r a l l a n g l e s 7 ? , . Summarizing and g e n e r a l i z i n g the r e s u l t s , f o r a sequence of homogeneous l a y e r s w i t h a c o m p r e s s i o n a l s o u r c e F ( t ) p l a c e d a t (r=0,z=0), o b s e r v a t i o n p o i n t P(r,z>0) and g i v e n a n g u l a r f r e q u e n c y o>, the r e p r e s e n t a t i o n of the t o t a l d i s p l a c e m e n t p o t e n t i a l at P i n terms of a s u p e r p o s i t i o n of c y l i n d r i c a l waves i s g i v e n .by (see 1-10 t o 1-13) * ( u , r , z ) = JV(w,k r ;z) [ 1 / i k\\ ]k r J 0 ( k r r ) d k r . (1-16) o where V((w,r ,z)} = J 3 / 3 z { V ( u , k r ; z ) } [ 1 / i k ; ] k r J 0 ( k r r ) d k r (1-17) o In e q u a t i o n (1-13) i t i s seen t h a t the z-dependence of V(w,k r;z) i s g i v e n o n l y by the terms e x p [ i i c j z c o s ( 7 ^ )/V p] = exp[±ik pz]. Hence (1-17) g i v e s S(o>,r,z) = /U(cj,k r ; z ) [ i k ; ][\\/ik\\ ] k r J 0 ( k r r )dk r (1-18) o and because of the ± s i g n s i n the argument of e x p f i k ^ z ] , U(o>,k r;z) and V ( c j , k r ; z ) may d i f f e r o n l y i n the s i g n a s s o c i a t e d t o each of t h e i r terms. I f the o b s e r v a t i o n p o i n t i s l o c a t e d a t the s u r f a c e z=0, e q u a t i o n (1-18) becomes S(w,r) = /U(aj,k r ) J 0 ( k r r ) k r d k r o (1-19) 1 5 Plane-wave d e c o m p o s i t i o n (PWD) i s u n d e r s t o o d as the problem of computing the plane-wave seismograms U(u>,kr) from the s p h e r i c a l - w a v e o b s e r v a t i o n s S ( o , r ) . M u l l e r (1971) r e c o g n i z e d e q u a t i o n (1-19) as a z e r o - o r d e r Hankel t r a n s f o r m and c o n s e q u e n t l y , p r e s e n t e d i t s f o r m a l i n v e r s i o n as U(cj,k r) = fs(cj, r ) J 0 ( k r r ) r d r (1-20) o p r o v i d i n g the b a s i c f o r m u l a t i o n f o r PWD. 1.2 On Plane-Wave D e c o m p o s i t i o n of D i g i t a l Data The i n t r o d u c t i o n of s l a n t s t a c k i n g ( S c h u l t z and C l a e r b o u t , 1978) and plane-wave d e c o m p o s i t i o n ( T r e i t e l e t a l . 1982) i n t o the r e a l m of r e f l e c t i o n s e i s m o l o g y has r e v e a l e d a l a r g e number of p o s s i b l e a p p l i c a t i o n s i n e x p l o r a t i o n s e i s m o l o g y . In a r e c e n t p u b l i c a t i o n , T r e i t e l e t a l . , (1982) have shown the r e l a t i o n s h i p s between s l a n t s t a c k i n g and plane-wave d e c o m p o s i t i o n . They p o i n t e d out t h a t a l t h o u g h e q u a t i o n (1-20) i s r e s t r i c t e d t o the v e r t i c a l component of c o m p r e s s i o n a l waves r e c o r d e d over l a t e r a l l y homogeneous media, t h i s f o r m a l i s m seems t o p r o v i d e ' an a c c e p t a b l e a p p r o x i m a t i o n of a l a r g e r c l a s s of e a r t h models, i n p a r t i c u l a r those c o n s i s t i n g of d i p p i n g l a y e r s . The g e n e r a l procedure f o l l o w e d t o p e r f o r m the plane-wave d e c o m p o s i t i o n of s p h e r i c a l - w a v e seismograms r e s u l t i n g from a 1 6 common-source g a t h e r i s i l l u s t r a t e d i n Table I . The a p p l i c a t i o n of PWD i n r e a l s e i s m i c work r e q u i r e s the d i s c r e t i z a t i o n of e q u a t i o n ( 1 - 2 0 ) . The form suggested by T r e i t e l e t a l . (1982) i s , U(w,k r) = ArLSdd.r. ) J 0 ( k r r - ) r f (1-21) In t h i s form, e q u a t i o n (1-21) may be a p p l i e d t o a s e t of seismograms S(o>,r; ) found a t e v e n l y spaced i n t e r v a l s Ar = r ;,, - r ; . In t h i s c a s e , the' i n t e g r a t i o n increment Ar i s f a c t o r e d out of the summation and i n t r o d u c e d l a t e r as a form of g l o b a l s c a l i n g . When uneven seismogram s p a c i n g i s e n c o u n t e r e d , we observe t h a t the f o r m a l use of e q u a t i o n (1-21) w i t h v a r i a b l e Ar f causes d e t e r i o r a t i o n of the d e c o m p o s i t i o n due t o i n a p p r o p r i a t e w e i g h t i n g of c e r t a i n terms i n the summation. In t h i s case i t might be b e t t e r t o e v a l u a t e (1-20) u s i n g an a p p r o p r i a t e n u m e r i c a l i n t e g r a t i o n scheme (e . g . i n t e r p o l a t i n g S(,kr) i s c o n s t r u c t e d . Henry e t a l . ' s s o l u t i o n i s e f f i c i e n t and i s a p p l i e d d i r e c t l y t o the f o r w a r d t r a n s f o r m a t i o n from t i m e - o f f s e t domain t o the d e l a y time - a n g l e of emergence domain ( i . e . from t - x t o r-y; because we a r e c o n c e r n e d w i t h one d i m e n s i o n a l e a r t h models, we make no d i s t i n c t i o n between the space v a r i a b l e s x and r ) . The same approach may be used f o r the i n v e r s e t r a n s f o r m a t i o n TABLE I 1 7 BASIC STEPS FOR PWD OF A COMMON-SOURCE GATHER COMMON-SOURCE GATHER Input of N s e i s m i c t r a c e s S ( t , r ; ) . FORWARD FOURIER TRANSFORM Temporal F o u r i e r t r a n s f o r m a t i o n of each t r a c e i n the common-source g a t h e r . C a l c u l a t i o n of S ( u u , r . - ) . PLANE-WAVE DECOMPOSITION At each a n g u l a r f r e q u e n c y u>, compu t a t i o n of U(UJ, k r ) (or e q u i v a l e n t l y of U(LU,JT)) f o r M d i f f e r e n t a n g l e s of emergence, INVERSE FOURIER TRANSFORM ( w i t h r e s p e c t to<^) From U ( L u , k r ) , c o m p u t a t i o n of M plane-wave seimograms U ( ^ , T ) . 18 ( i . e . from the r-y domain t o the t-x domain). S i n c e f o r some a p p l i c a t i o n s PWD may ser v e as a f i l t e r i n g o p e r a t i o n , our o b j e c t i v e i s t o modify Henry et a l . ' s s o l u t i o n and produce a more f l e x i b l e and s t a b l e a l g o r i t h m which w i l l p e r f o r m both the fo r w a r d ( t - x t o r-y) and the i n v e r s e (r-y t o t- x ) t r a n s f o r m a t i o n s . Rather than use the i n n e r p r o d u c t g i v e n by Henry et a l . , we i n t r o d u c e an e x p l i c i t w e i g h t i n g f u n c t i o n i n t o ( 1 - 1 9 ) OO S ( u , r ) = /[U(w,k r )Q\" 1 ] [ Q J 0 U R r ) k r ]dk r , o f o r the f o r w a r d t r a n s f o r m a t i o n , and i n t o (1-20) U ( u , k r ) = / [ S ( u , r ) Q - 1 ] [ Q J 0 U . r ) r ] d r o f o r the i n v e r s e t r a n s f o r m a t i o n . T h i s i s an example of a l i n e a r q u e l l i n g (Backus, 1970), and a l l o w s the use of the u s u a l d e f i n i t i o n of the i n n e r p r o d u c t of two f u n c t i o n s . W i t h t h i s , i n a d d i t i o n t o f i n d the s m a l l e s t model s o l u t i o n s f o r U(co,k r)Q\" 1 and f o r S ( c j , r ) Q \" ' , we f i n d the f l a t t e s t model s o l u t i o n s . F i n a l l y , the s o l u t i o n s of the f o r w a r d and i n v e r s e t r a n s f o r m a t i o n s a r e found s u b j e c t t o the x 2 c r i t e r i o n (see Appendix A ) , so t h a t o b s e r v a t i o n a l e r r o r s a r e a c c o u n t e d f o r . The a l g o r i t h m s a re a p p l i e d t o the problems of s e p a r a t i n g c o n v e r t e d S modes from o t h e r modes i n a common-source g a t h e r , 19 and of t r a c e i n t e r p o l a t i o n . In the f i r s t problem we w i l l f o l l o w Tatham et a l . (1983), and t r a n s f o r m the common-source g a t h e r ( t - x domain) t o the plane-wave domain (r-y domain). For reasons which w i l l be o u t l i n e d l a t e r , c e r t a i n c o n v e r t e d modes w i l l occupy a d i s t i n c t p o r t i o n of the plane-wave domain. I n v e r s e t r a n s f o r m a t i o n of o n l y t h i s p o r t i o n back t o t-x space w i l l y i e l d t he common-source g a t h e r (CSG) r e p r e s e n t a t i o n of the S modes p r e s e n t i n the chosen r-y zone. The second problem i s s o l v e d by u t i l i z i n g the a l g o r i t h m s t o c o n s t r u c t a d d i t i o n a l seismograms a t o f f s e t s not r e p r e s e n t e d i n the o r i g i n a l CSG. I t i s i m p o r t a n t t o emphasize t h a t a l t h o u g h the use of ( 1 - 1 9 ) and (1-20) are r e s t r i c t e d t o the r e c o r d e d c o m p r e s s i o n a l waves a t the s u r f a c e z=0, we r e a l i z e t h a t k i n e m a t i c a l l y t h e s e e q u a t i o n s a r e s t i l l s a t i s f a c t o r y f o r o b t a i n i n g the plane-wave s i g n a t u r e of r e c o r d e d S waves. The reason f o r t h i s i s u n d e r s t a n d a b l e from the d i s c u s s i o n d e v e l o p i n g ( 1 - 1 2 ) . In the case of S p l a n e waves, V(u>,k r;z) has v e r t i c a l p r o p a g a t i o n terms of t he form exp[ ic j z c o s ( 7 s , )/V* ] and hence, the d e l a y times a r e s t i l l governed by c o s i n e f u n c t i o n s . In Chapter 3 we w i l l r e t u r n t o t h i s m a t t e r . 20 2. BACKUS AND GILBERT FORMULATION OF THE PROBLEM 2.1 Forward Transform (t-x to r-y): C o n s t r u c t i o n of the Plane-Wave Components of Spherical-Wave Seismograms To apply the Ba c k u s - G i l b e r t (E-G) theory to the problem of plane-wave decompositon, we use equation (1-19) at s p e c i f i e d o f f s e t r- and angular frequency u, that i s S(w,r,- ) = /U(cj,k r ) J 0 ( k r r ; ) k r d k r ( 2 - 1 ) o We can now solve ( 2 - 1 ) f o r U(cj,k r) as a set of i n v e r s e problems each of which corresponds to a given angular frequency co. In order to expedite the f o l l o w i n g p r e s e n t a t i o n we introduce the terminology and n o t a t i o n s to be used throughout the remainder of t h i s work: (a) S(cj,r : ), the temporal F o u r i e r - t r a n s f o r m e d elements of the spherical-wave seismograms, at a given angular frequency CJ and o f f s e t r- , are termed 'observations' and are denoted by e?. (b) U(w,k r), the temporal F o u r i e r transform (FT) of the plane-wave seismograms at a given angular frequency u>, i s termed the 'model' and i s denoted by m; m i s a continuous f u n c t i o n of the h o r i z o n t a l wave number k r.. 21 (c) J 0 ( k r r ; ) k r , the z e r o - o r d e r B e s s e l f u n c t i o n s m u l t i p l i e d by the h o r i z o n t a l wave number, a r e termed ' k e r n e l s ' and a r e denoted by G;. They are c o n t i n u o u s f u n c t i o n s of the h o r i z o n t a l wave number k r . (d) The i n n e r product of the f u n c t i o n s f ( k ) and g(k) i s CO denoted by ( i . e . , = / f ( k ) g ( k ) d k ). o In the f o l l o w i n g two s e c t i o n s we w i l l o u t l i n e the B-G s o l u t i o n t o problems of the form of e q u a t i o n (2-1) t o show how s m a l l e s t and f l a t t e s t models can be computed. F u r t h e r t r e a t m e n t of the proce d u r e i s found i n P a r k e r (1977), and Oldenburg and Samson (1979). 2.1.1 S m a l l e s t model c o n s t r u c t i o n ( f o r w a r d t r a n s f o r m ) C o n s i d e r the problem, e? = i = 1 , . . . ,N (2-2) where N i s the number of o b s e r v a t i o n s . Assume t h a t the g i v e n o b s e r v a t i o n s a r e c o n t a m i n a t e d by e r r o r s {5e ;} w i t h z e r o mean and e s t i m a t e d s t a n d a r d d e v i a t i o n a,-, i . e . e? = et±6e,- , e* b e i n g the t r u e d a t a . T h e r e f o r e the e q u a t i o n t o be s o l v e d i s , e*±5e,- = which upon d i v i s i o n by a- becomes, [e;*±6e; ]/o; = 22 e? = (2-3) e° a r e our new o b s e r v a t i o n s w i t h a s s o c i a t e d e r r o r s of u n i t v a r i a n c e , and G ; a r e the new s c a l e d k e r n e l s . G i v e n N o b s e r v a t i o n s e? and the f u n c t i o n a l form ( 2 - 3 ) , we would l i k e t o o b t a i n a c o n t i n u o u s model m. T h i s problem i s always underdetermined and admits i n f i n i t e l y many s o l u t i o n s but a s p e c i f i c model i s o b t a i n e d by m i n i m i z i n g some norm of the model and u s i n g the d a t a e q u a t i o n s as c o n s t r a i n t s . The s m a l l e s t model c o r r e s p o n d s t o the requirement t h a t the L 2 norm of the c o n s t r u c t e d model w i l l be s m a l l e r than t h a t of any o t h e r p e r m i s s i b l e s o l u t i o n ( i . e . a l l those s a t i s f y i n g ( 2 - 3 ) ) . G i v e n t h i s r e q u i r e m e n t , the s o l u t i o n t o (2-3) can be e x p r e s s e d as a l i n e a r c o m b i n a t i o n of the k e r n e l s , t h a t i s (Oldenburg and Samson, 1979), The c o e f f i c i e n t s a f a r e o b t a i n e d by s u b s t i t u t i n g (2-4) i n t o ( 2 - 3 ) , c h a n g i n g the o r d e r of summation and i n t e g r a t i o n and s o l v i n g the system, m = I a : G ; (2-4) e° = Ta (2-5a) t h a t i s , a = T- 1e° (2-5b) where a i s the v e c t o r of k e r n e l c o e f f i c i e n t s , e° i s the v e c t o r of o b s e r v a t i o n s , and 23 T~1 i s the i n v e r s e of the (NxN) i n n e r p r o d u c t m a t r i x T d e f i n e d by, r,-= (2-6) The f o r m a l s o l u t i o n g i v e n i n e q u a t i o n s (2-4) t o (2-6) cannot y i e l d a p h y s i c a l s o l u t i o n i f the k e r n e l s G,- are not square i n t e g r a b l e . The c u r r e n t problem of PWD i s an example of t h i s o c c u r r e n c e . A way to c i r c u m v e n t t h i s i s t o use the q u e l l i n g o p e r a t i o n (Backus,1970) which i s e s s e n t i a l l y a mapping of the k e r n e l s i n t o a H i l b e r t space. The method we use i s named \" q u e l l i n g by m u l t i p l i c a t i o n \" (Backus,1970) i n which we lo o k f o r a w e i g h t i n g f u n c t i o n Q such t h a t G? = G,-Q i s i n L2(0,°°) f o r a l l i . Once such a Q i s s p e c i f i e d we r e w r i t e (2-3) a s , e? = = (2-7) and c o n t i n u e t o f i n d the s m a l l e s t model m* as o u t l i n e d i n e q u a t i o n s (2-4) t o ( 2 - 6 ) . S u b s e q u e n t l y we \"unweight\" m* and o b t a i n the d e s i r e d model. The f i n a l s o l u t i o n i s then g i v e n by, m = La; Q 2G ; (2-8) An i m p o r t a n t c o n s i d e r a t i o n i n the c h o i c e of the f u n c t i o n Q i s the ease w i t h which the e v a l u a t i o n of the i n n e r p r o d u c t r;j-= can procee d . The e f f i c i e n c y of the c o n s t r u c t i o n a l g o r i t h m i n c r e a s e s g r e a t l y i f an a n a l y t i c a l e x p r e s s i o n r e p r e s e n t i n g the elements of the i n n e r p r o d u c t m a t r i x \" i s found. A l s o , the w e i g h t i n g f u n c t i o n Q s h o u l d l e a d t o an e f f i c i e n t 24 c o n s t r u c t i o n of the i n v e r s e t r a n s f o r m ( i . e . from 7 - 7 t o t - x ) . Indeed, s i n c e many i n v e r s e s o l u t i o n s are r e q u i r e d (one f o r each f r e q u e n c y ) , n u m e r i c a l e f f i c i e n c y i s g a i n e d i f Q i s chosen so t h a t o n l y a s i n g l e m a t r i x s p e c t r a l d e c o m p o s i t i o n i s r e q u i r e d . I f t h i s o b j e c t i v e can be a c h i e v e d , the i n t r o d u c t i o n of the x 2 c r i t e r i o n (Appendix A) f o r n o i s y d a t a does not decrease the a l g o r i t h m ' s e f f i c i e n c y . Our s o l u t i o n t o the s m a l l e s t - m o d e l f o r w a r d c o n s t r u c t i o n i n v o l v e s the w e i g h t i n g f u n c t i o n Q = k r \" °-5Kc0-5 (k r b) , where K 0 i s a m o d i f i e d B e s s e l f u n c t i o n of z e r o o r d e r , and b i s an a r b i t r a r y p o s i t i v e r e a l number whose r o l e i s demonstrated i n Appendix B. Wit h the above c h o i c e of w e i g h t i n g the c o n s t r u c t i o n proceeds w i t h t he f o l l o w i n g i d e n t i f i c a t i o n s , nr = m/Q = k r °-5K0-°-s ( k r b ) U ( c j , k r ) (2-9a) and Gr = GrQ = k r °-5K00-5 (k r b) J 0 ( k r r : )/o; (2-9b) Hence, from e q u a t i o n (2-8) the frequ e n c y \" r e p r e s e n t a t i o n of the plane-wave seismograms i s g i v e n by, U( i s d e s c r i b e d i n Appendix C, whereas d e t e r m i n a t i o n of the c o e f f i c i e n t s a ; t o o b t a i n p r o p e r x 2 v a l u e i s d i s c u s s e d i n Appendix A. 2.1.2 F l a t t e s t model c o n s t r u c t i o n ( f o r w a r d t r a n s f o r m ) C o n s i d e r the problem o u t l i n e d i n the p r e v i o u s s e c t i o n , i . e . g i v e n N o b s e r v a t i o n s c o r r e s p o n d i n g t o N f u n c t i o n a l r e l a t i o n s e° = , f i n d a model m which s a t i s f i e s these r e l a t i o n s - In t h i s s e c t i o n , we s e a r c h f o r the model which e x h i b i t s the l e a s t amount of change w i t h r e s p e c t t o the independent v a r i a b l e . The c o n s t r u c t i o n of t h i s type of model (commonly r e f e r r e d t o as the f l a t t e s t model) i s a c h i e v e d by the m i n i m i z a t i o n of the norm ||m'||, w i t h m' b e i n g the d e r i v a t i v e of the model. In the problem of PWD, the f l a t t e s t model r e q u i r e s t h a t the F o u r i e r t r a n s f o r m of the PWS e x h i b i t s the l e a s t amount of v a r i a t i o n w i t h r e s p e c t t o k r a t each a n g u l a r f r e q u e n c y o>, and hence i t may y i e l d b e t t e r l a t e r a l c o n t i n u i t y i n terms of both a m p l i t u d e and time d e l a y . In o r d e r t o c o n s t r u c t the f l a t t e s t model we i n t e g r a t e the r . h . s . of e q u a t i o n (2-3) by p a r t s t o o b t a i n , OO e? = H,- m| - (2-1 1 ) o where, H ; ( k r ) = /G;(u)du 2 6 Presuming we can evaluate the term R;m| we s u b s t r a c t i t from the o l e f t hand s i d e to get, e,t = - (2-12) Using the technique o u t l i n e d i n s e c t i o n 2.1.1 above we proceed to f i n d the smallest m' model. If the new kernels H; are not square i n t e g r a b l e , we intr o d u c e a weighting f u n c t i o n Q and then s o l v e e;t = <-H;Q,m'/Q> to ob t a i n m'. The s o l u t i o n to t h i s problem i s given by ( c f . equation 2 - 8 ) , m' = -Z/3- Q2H,- (2-13a) I - r where the 0,- are obtained from, I = r - ' l t (2-13b) and, r..3 = (2-13c) Taking the i n d e f i n i t e i n t e g r a l of equation (2-l3a) we o b t a i n , m(k r) = -Z/3j JQ 2 (u)R,- (u)du + C (2-14) Two important c o n s i d e r a t i o n s should be emphasized at t h i s po i n t : (a) the c o n s t r u c t i o n of the f l a t t e s t model n e c e s s i t a t e s the a d d i t i o n a l knowledge of a (boundary) value of m, from which the constant C i s found. 27 (b) the c h o i c e of the w e i g h t i n g f u n c t i o n Q i s now burdened by the a d d i t i o n a l e v a l u a t i o n of the i n d e f i n i t e i n t e g r a l JH.Q 2. In the s o l u t i o n t o the f o r w a r d - t r a n s f o r m f l a t t e s t - m o d e l c o n s t r u c t i o n we make the f o l l o w i n g i d e n t i f i c a t i o n s : (a) G ; = k r J 0 ( k r r : )/ot . (b) The new k e r n e l s -H;= - k r J , ( k r r , ) / [ o ; r ; ] (see Appendix D). (c) L i m i t i n g r ; t o be g r e a t e r than z e r o , we have H ;m|=0. On o the o t h e r hand, assuming a band l i m i t e d s o u r c e f u n c t i o n , kr-*<» i m p l i e s i n f i n i t e l y - a t t e n u a t e d inhomogeneous waves. Hence we c o n s i d e r H;m| =0 (see Appendix D), which means t h a t the new o b s e r v a t i o n s e-f a r e the same as e f . (d) The w e i g h t i n g f u n c t i o n Q we have chosen i s K,°-5(krb) where i s a m o d i f i e d B e s s e l f u n c t i o n of f i r s t o r d e r and b i s an a r b i t r a r y p o s i t i v e r e a l number (see Appendix B ) . We n o t i c e t h a t because k r=wsin ( 7)/V, a d i f f e r e n t w e i g h t i n g f u n c t i o n Q i s used f o r each a n g u l a r f r e q u e n c y u>. (e) The c o n s t a n t of i n t e g r a t i o n C i s e q u a l t o zer o (see Appendix D). F o l l o w i n g the development i n Appendix D, t h e plane-wave seismograms a t a g i v e n a n g u l a r f r e q u e n c y a re t h e r e f o r e g i v e n by, U ( u , k r ) = -L0; JQ2H,- = |[0j /o; r , ] { b k r J , (k r r ; ) K 0 ( k r b ) + r. k r J 0 (k rr,- )K, ( k r b ) } / [ r ; 2 + b 2 ] (2-15) 28 2.2 I n v e r s e T ransform (r-y t o t - x ) : R e c o n s t r u c t i o n of the S p herical-Wave F i e l d from i t s Plane-Wave Components The i n v e r s e problem t o be s o l v e d i s e x p r e s s e d i n e q u a t i o n ( 1 - 2 0 ) . k r has been r e p l a c e d by r as the independent v a r i a b l e , and the model and o b s e r v a t i o n s have t r a d e d p l a c e s so t h a t the former now r e p r e s e n t s the t e m p o r a l FT of the s p h e r i c a l - w a v e seismograms S(w,r) whereas the l a t t e r c o n s i s t s of the FT of the plane-wave seismograms U(&>, k r ).. Indeed, t h e r e i s no d i f f e r e n c e between c o n s t r u c t i n g plane-wave and s p h e r i c a l - w a v e seismograms from each o t h e r i n the way f o r m u l a t e d i n the p r e v i o u s s e c t i o n . However, because we have f o r m u l a t e d the problem i n terms of the h o r i z o n t a l wavenumber k r r a t h e r than i n terms of the ray parameter p, the i n n e r p r o d u c t m a t r i x f o r the i n v e r s e t r a n s f o r m a t i o n e x p l i c i t l y depends on the a n g u l a r f r e q u e n c y (see Appendix E ) . To see t h i s and compare the form of the i n n e r p r o d u c t m a t r i c e s f o r the f o r w a r d and i n v e r s e t r a n s f o r m a t i o n s , l e t us c o n s i d e r the f o l l o w i n g i n t e g r a l s r;F- = j G ; ( k r r ; )G- (k rr_; ) Q 2 ( k r b ) d k r (2-16a) o and T-j = ; G ; ( k r . r ) G j ( k f j r ) Q 2 ( b r ) d r . (2-16b) r;F- i s an element of the i n n e r p r o duct m a t r i x f o r the f o r w a r d t r a n s f o r m a t i o n w h i l e r,* i s an element f o r the i n v e r s e t r a n s f o r m a t i o n . In terms of the ray parameter p, (2-16a) and (2-!6b) r e a d as 29 f oo r.- = /G; (upr ; )Gj (cjprj )Q 2 (upb)dup (2-17a) o and r?i = o/G; (up. r)G- (up^rjQMbrJdr. ( 2 - l 7 b ) These integrals have the same form. I f in ( 2 - l 7 a ) we set b=r c =positive constant, and in (2-17b) b=cjpc , with p„=positive constant we find rf- = /G; (cjpr ; )G0 (cjprj )Q 2 ( u p r c )dwp (2-18a) and r?j = 1/CJ /G; (£jp; r ) G j ( c j p J r ) Q 2 ( u p c r ) d c j r . ( 2 - l 8 b ) T h e r e f o r e , integration of ( 2 - l 8 a ) with respect to cjp and ( 2 - l 8 b ) with respect to ur w i l l give matrices r* and r A which can have m u l t i p l i c a t i v e factors dependent on u. In both transformations, spectral decomposition of a single inner product matrix i s done only once. For the inverse transformation the d e f i n i t i o n of b=a>pc in the argument of Q means that, as in the forward problem, a di f f e r e n t weighting function is used at each angular frequency CO. 30 2.2.1 S m a l l e s t model c o n s t r u c t i o n ( i n v e r s e t r a n s f o r m ) The proc e d u r e here i s p a r a l l e l t o t h a t o u t l i n e d i n the s e c t i o n d e a l i n g w i t h the c o r r e s p o n d i n g f o r w a r d t r a n s f o r m . I d e n t i f y i n g , G; = r J 0 ( r k r ; )/a ; Q = r-°-5K0°-s ( r b ) and u s i n g e q u a t i o n ( 2 - 8 ) , the s m a l l e s t model f o r the s p h e r i c a l -wave seismograms a t a g i v e n u> i s , S(w,r) = Z [ a ; / c ; ] K 0 ( rb) J 0 ( r k r ; ) w i t h b = w s i n ( c ) / V (2-19) where M i s the number of plane-wave seismograms. 2.2.2 F l a t t e s t model c o n s t r u c t i o n ( i n v e r s e t r a n s f o r m ) The development here i s s i m i l a r t o t h a t of the c o r r e s p o n d i n g f o r w a r d t r a n s f o r m . At the st a g e of the s o l u t i o n of the s m a l l e s t model m', the s u b s t i t u t i o n b = c j s i n ( c ) / V i s made (Appendix E ) . Assuming S (a>, r=°°) =0, the i n t e g r a t i o n c o n s t a n t C i n e q u a t i o n (2-14) i s z e r o . Hence the freq u e n c y r e p r e s e n t a t i o n of the v e r t i c a l component of the s p h e r i c a l - w a v e seismograms i s g i v e n by, 31 b = S ( u , r ) = Z[/3,/a, k r ; ] { b r J , ( k r ; r ) K 0 ( r b ) i - t + k r ; r J 0 ( k r ; r ) K 1 ( r b ) } / [ k 2+b 2 ]} us i n ( c ) / V ( 2 - 2 0 ) where M i s the number of plane-wave seismograms, 32 3. EXAMPLES 3.1 I n t r o d u c t o r y Comments In the f o r w a r d t r a n s f o r m our g o a l i s t o o b t a i n plane-wave seismograms U(r,y) from s p h e r i c a l - w a v e seimograms S ( t , x ) . To a c h i e v e t h i s , we f o l l o w the s t e p s i l l u s t r a t e d i n T a b l e I of s e c t i o n 1.2 (p. 17). C o n v e r s e l y , i n the i n v e r s e t r a n s f o r m our o b j e c t i v e i s t o compute the v e r t i c a l - d i s p l a c e m e n t s p h e r i c a l - w a v e seismograms S ( t , x ) from plane-wave seismograms U{r,y). To a c h i e v e t h i s we s t i l l f o l l o w those s t e p s g i v e n i n T a b l e I though kee p i n g i n mind t h a t the i n p u t d a t a are plane-wave seismograms U ( r , 7 ) . We r e f e r t o f o r m a t i o n of the plane-wave or s p h e r i c a l -wave seismograms v i a the d i s c r e t e form of e q u a t i o n (1-20) or e q u a t i o n (1—19) as the Hankel t r a n s f o r m , and c o n s t r u c t i o n i n v o l v i n g the B-G i n v e r s i o n as the s m a l l e s t or f l a t t e s t model. In t he examples t o f o l l o w we w i l l o b t a i n plane-wave seismograms f o r v a l u e s of the a n g l e of emergence y between 0° and 90° ( t h a t i s t o say 0>1 ( i . e . f o r h i g h f r e q u e n c i e s and/or l a r g e o f f s e t s ; A k i and R i c h a r d s , 1980, c h . 6 ) . In (a) c l o s e r e c e i v e r s c o n t r i b u t e t o form plane-wave s i g n a l a t s m a l l a n g l e s of emergence whereas f a r r e c e i v e r s do i t f o r l a r g e emergence a n g l e s . N o t i c e the d i f f e r e n c e s i n d e n s i t y of a n g u l a r i n f o r m a t i o n f o r s m a l l and l a r g e o f f s e t s . In (b) the s p h e r i c a l - w a v e s i g n a l from a deep i n t e r f a c e has s m a l l moveout throughout the r e c e i v e r s and hence, i t s plane-wave si-gna-1 w i l l be o b s e r v e d a t s m a l l a n g l e s of emergence. 41 of the o b s e r v e d modes i n the plane-wave domain, we d i s p l a y i n F i g u r e 3.5 the t h e o r e t i c a l t r a j e c t o r i e s of the modes which a r e e x p e c t e d f o r the g i v e n model. To s t a r t t h i s example, we show the Hankel plane-wave seismograms i n F i g u r e 3.6. The c o n v e r t e d S modes PPSS, PSSP, PSSS and PS a r e seen q u i t e c l e a r l y i n the a r e a 0.2S 2. 2 m n 2. 4 2. 6 2. 8 3. 0 3. 2 3. 4 3 . 6 3 . 8 4. 0 FIGURE 3.21 C o n v e r t e d S-mode t - x seismograms o b t a i n e d by ( i ) PWD of F i g u r e 3.19 u s i n g the f o r w a r d s m a l l e s t - m o d e l a l g o r i t h m , and ( i i ) r e c o n s t r u c t i o n of the t - x seismograms u s i n g the i n v e r s e f l a t t e s t - m o d e l a l g o r i t h m . The seismograms a r e n o r m a l i z e d by t r a c e . 66 4. COMPUTATIONAL CONSIDERATIONS The CPU t i m e s (on an Amdahl 470 V/8 computer) f o r the PWD a l g o r i t h m s are p r e s e n t e d i n Table I I below. The seismograms used to g e n e r a t e t h i s t a b l e were d i g i t i z e d a t 8 ms and were l i m i t e d t o the band . 5-25 Hz (the number of frequency samples i n t h i s band i s e q u i v a l e n t t o those c o n t a i n e d i n the band 10-50 Hz on data w i t h 4 ms s a m p l i n g i n t e r v a l ) . F u r t h e r m o r e , a l l c a l c u l a t i o n s ( w i t h the e x c e p t i o n of B e s s e l f u n c t i o n c o m p u t a t i o n s ) were made i n double p r e c i s i o n . TABLE I I TIME INPUT OUTPUT CPU TIMES SAMPLES TRACES TRACES HANKEL SMALLEST FLATTEST 1 28 256 512 . 40 40 40 45 45 45 2.416s 4.768s 9.349s 3 .556s 6.770s 12.965s 5.342s 10.192s 19.909s The f o l l o w i n g p o i n t s a r e emphasized: ( i ) Most of the p r o c e s s i n g time i s e l a p s e d i n B e s s e l f u n c t i o n c o m p u t a t i o n s . For t h e s e , we have used p o l y n o m i a l a p p r o x i m a t i o n s w i t h e r r o r s of 0 ( 1 0 \" 8 ) g i v e n i n Abramowitz and Stegun (1970). E f f i c i e n c y of t h e s e c o m p u t a t i o n s can be i n c r e a s e d by: (a) d e c r e a s i n g the a c c u r a c y of the a p p r o x i m a t i o n s which w i l l reduce p r o c e s s i n g t i m e s , though i t 67 may y i e l d some d e g r a d a t i o n of the o u t p u t , and (b) u s i n g the d e r i v a t i v e r e l a t i o n between J 0 and J , i n f l a t t e s t model c a l c u l a t i o n s . From the summation r e p r e s e n t e d by e q u a t i o n (2-10) f o r the s m a l l e s t or t h a t of e q u a t i o n (2-15) f o r the f l a t t e s t model, i t i s deduced t h a t i n c r e a s i n g the number of i n p u t or output t r a c e s causes a l i n e a r i n c r e a s e i n the number of B e s s e l f u n c t i o n c o m p u t a t i o n s . S i m i l a r l y , an i n c r e a s e i n the number of time or frequency samples r e s u l t s i n a l i n e a r i n c r e a s e i n the number of B e s s e l f u n c t i o n c o m p u t a t i o n s . ( i i ) The CPU time f o r the s i n g u l a r v a l u e d e c o m p o s i t i o n (SVD) of the i n n e r p r o d u c t m a t r i x T behaves l i k e the cube of the number of i n p u t t r a c e s . Hence, depending on the r a t i o between the number of time and freq u e n c y samples t o the number of i n p u t t r a c e s , a s i g n i f i c a n t p e r c e n t a g e of the t o t a l run time may be spent i n SVD. However, when p r o c e s s i n g a l a r g e number of CSG's w i t h f i x e d geometry and s t a n d a r d d e v i a t i o n e s t i m a t e s , SVD i s ex e c u t e d once. In t h i s c a s e , the t o t a l s m a l l e s t - m o d e l CPU time i s comparable t o t h a t of the Hankel a l g o r i t h m . ( i i i ) The memory r e q u i r e m e n t s of the B-G a l g o r i t h m s i n c l u d e one a r r a y of s i z e NPTSxNTRACE and two of s i z e NTRACExNTRACE, where NTRACE i s the number of e i t h e r i n p u t or o u t p u t t r a c e s w h i c h e v e r i s l a r g e r , and NPTS i s the number of samples per t r a c e . In c o n t r a s t , the Hankel a l g o r i t h m needs o n l y one a r r a y of s i z e NPTSxNTRACE. 68 ( i v ) The CPU run t i m e s and memory re q u i r e m e n t s c o r r e s p o n d i n g t o the i n v e r s e t r a n s f o r m a l g o r i t h m s are e q u i v a l e n t t o those of the f o r w a r d t r a n s f o r m . 69 5. SUMMARY E f f i c i e n t a l g o r i t h m s f o r the d e c o m p o s i t i o n of a s p h e r i c a l -wave f i e l d i n t o i t s plane-wave components have been p r e s e n t e d . A l s o , i t has been shown t h a t t h e s e a l g o r i t h m s a l l o w the r e c o n s t r u c t i o n of the s p h e r i c a l - w a v e f i e l d from a r e l a t i v e l y s p a r s e sample of i t s plane-wave components. The p r a c t i c a l v i a b i l i t y of the proposed a l g o r i t h m s has been demonstrated u s i n g the problem of s e p a r a t i o n of c o n v e r t e d S modes from o t h e r modes i n a common-source g a t h e r , and the problem of t r a c e i n t e r p o l a t i o n . The f o l l o w i n g p o i n t s s h o u l d be n o t e d : ( i ) The plane-wave seismograms a r e o b t a i n e d by u s i n g the Backus & G i l b e r t c o n s t r u c t i o n t e c h n i q u e s , s u b j e c t e d t o the requirement of w e i g h t e d s m a l l e s t or f l a t t e s t model. ( i i ) The c o n s t r u c t i o n schemes a l l o w the h a n d l i n g of e r r o r s i n the d a t a and hence, p e r m i t a c e r t a i n c o n t r o l on the model s t r u c t u r e p r o v i d e d by the b a s i s f u n c t i o n s . C a u t i o n s h o u l d be e x e r c i s e d i n a s s i g n i n g the s t a n d a r d d e v i a t i o n v a l u e s t o u n n o r m a l i z e d o b s e r v a t i o n s . ( i i i ) The proposed a l g o r i t h m s a r e not l i m i t e d t o e v e n l y spaced d a t a and c o n s e q u e n t l y , a l l o w the d e s i g n of an a p p r o p r i a t e geophone a r r a y which s h o u l d produce a more f a i t h f u l r e p r e s e n t a t i o n of the plane-wave components. 70 ( i v ) N u m e r i c a l s t a b i l i t y i s g a i n e d by proper use of the b v a l u e a p p e a r i n g i n the arguments of the w e i g h t i n g f u n c t i o n s . Large b v a l u e s decrease the degree of l i n e a r independence of the k e r n e l s and seem t o be a p p r o p r i a t e f o r the problem of t r a c e i n t e r p o l a t i o n . (v) Dynamic a s p e c t s ( e.g: t r u e a m p l i t u d e s ) of the forwa r d and i n v e r s e c o n t r u c t e d models are not f o r m a l l y handled by the a l g o r i t h m s as d e v e l o p e d . ( v i ) For a g i v e n a n g l e of emergence 7 , homogeneous p l a n e waves c o r r e s p o n d i n g to h i g h f r e q u e n c i e s may be s i g n i f i c a n t l y a t t e n u a t e d by the w e i g h t i n g f u n c t i o n Q. T h i s f u n c t i o n t h e n , i s viewed as a p o t e n t i a l a l i a s i n g s u p p r e s s o r . ( v i i ) For a g i v e n a n g u l a r frequency C J , l a r g e wavenumber components are s e v e r e l y a t t e n u a t e d by the w e i g h t i n g f u n c t i o n Q. Indeed, depending on the b v a l u e chosen, inhomogeneous waves a s s o c i a t e d w i t h wavenumbers l a r g e r than a c e r t a i n v a l u e a re p r a c t i c a l l y e x c l u d e d from the d e c o m p o s i t i o n . T h i s e f f e c t i s analogous t o f o r m u l a t i n g PWD as an i n v e r s e problem w i t h f i n i t e l i m i t s of i n t e g r a t i o n . 71 BIBLIOGRAPHY Abramowitz, M. , and Stegun, I . , 1970, Handbook of M a t h e m a t i c a l F u n c t i o n s : New York, Dover P u b l i c a t i o n s I n c . , 1046 p. A k i , K., and R i c h a r d s , P. G., 1980, Q u a n t i t a t i v e Seismology. Theory and Methods: San F r a n c i s c o , W. H. Freeman and Co., v. I , 557 p. Asby, R., and Wolf, E., 1971, Evanescent waves and the' e l e c t r o m a g n e t i c f i e l d of a moving charged p a r t i c l e : J . Opt. Soc. Am., v. 61, p. 52-59. Backus, G., 1970, I n f e r e n c e from inadequate and i n a c c u r a t e d a t a , I I : P r o c . Nat. 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Fuchs, K., and M u l l e r , G., 1971, Computation of s y n t h e t i c seismograms w i t h the r e f l e c t i v i t y method and c o m p a r i s i o n w i t h o b s e r v a t i o n s : Geophys. J . R. A s t r . S o c , v. 23, p. 417-433. Goodman, J . W. , 1968, I n t r o d u c t i o n t o F o u r i e r O p t i c s : San F r a n c i s c o , M c G r a w - H i l l Co., 287 p. G r a d s h t e y n , I . , and R y z h i k , I . , 1980, T a b l e of I n t e g r a l s , S e r i e s and P r o d u c t s : T o r o n t o , Academic P r e s s , 1160 p. Henry, M., O r c u t t , J . , and P a r k e r , R., 1980, A new method f o r s l a n t s t a c k i n g r e f r a c t i o n d a t a : Geophys. Res. L e t t . , v. 7, p. 1073-1076. 72 M u l l e r , G., 1971, D i r e c t i n v e r s i o n of s e i s m i c o b s e r v a t i o n s : Z e i t s c h r i f t f u r Geophysik, v. 37, p. 225-235. Oldenburg, D. W, and Samson, J . C, 1979, I n v e r s i o n of i n t e r f e r o m e t r i c d a t a from c y l i n d r i c a l l y symmetric r e f r a c t i o n l e s s plasmas: J . Opt. Soc. Am., v. 69, p. 927-941. P a r k e r , R. L., 1977, U n d e r s t a n d i n g i n v e r s e t h e o r y : Ann. Rev. E a r t h P l a n e t . S c i . , v. 5, p. 35-64. Ryu, J . V., 1982, D e c o m p o s i t i o n (DECOM) approach a p p l i e d t o wave f i e l d a n a l y s i s w i t h s e i s m i c r e f l e c t i o n r e c o r d s : G e o p h y s i c s , v. 47, p. 869-883. S c h u l t z , P. S., and C l a e r b o u t , J . F., 1978, V e l o c i t y e s t i m a t i o n and downward c o n t i n u a t i o n by wavefront s y n t h e s i s : G e o p h y s i c s , v. 43, p. 691-714. S t r a t t o n , J . A., 1941, E l e c t r o m a g n e t i c Theory: New York, McGraw-H i l l Co., 615 p. Tatham, R. H., Goolsbee, D. V., M a r s e l , W. F., and N e l s o n , H. R., 1983, S e i s m i c shear-wave o b s e r v a t i o n s i n a p h y s i c a l model ex p e r i m e n t : G e o p h y s i c s , v. 48, p. 688-701. T r e i t e l , S., G u t o w s k i , P., and Wagner, D., 1982, Plane-wave d e c o m p o s i t i o n of seismograms: G e o p h y s i c s , v. 47, p. 1375-1401. 73 APPENDIX A X 2 V a l u e and O b s e r v a t i o n a l E r r o r s i n Model C o n s t r u c t i o n When s o l v i n g problems which a re a s s o c i a t e d w i t h i n a c c u r a t e o b s e r v a t i o n s , i t i s u n d e s i r a b l e t o c o n s t r u c t models which reproduce these d a t a e x a c t l y . In t h i s c a s e , i t i s common t o r e q u i r e the c a l c u l a t e d o b s e r v a t i o n s t o f i t the d a t a i n a manner c o n s i s t e n t w i t h the o b s e r v a t i o n a l e r r o r s . In t h i s appendix, we o u t l i n e the s t e p s r e q u i r e d i n the c o n s t r u c t i o n of models w i t h c a l c u l a t e d o b s e r v a t i o n s which a r e r e l a t e d t o the observed d a t a by an ex p e c t e d x 2 v a l u e of a p p r o x i m a t e l y N. F i r s t l y , from e q u a t i o n (2-5a) we have, Ta = 1° E x p r e s s i n g T i n terms of i t s s p e c t r a l components and s o l v i n g f o r a we g e t , a = r- 1t° = RA\" 1R Te° (A-1) where R i s an (NxN) m a t r i x whose columns a re the e i g e n v e c t o r s of T, A i s an (NxN) d i a g o n a l m a t r i x whose d i a g o n a l c o n s i s t s of the e i g e n v a l u e s of T a r r a n g e d i n d e c r e a s i n g o r d e r , and R T i s the t r a n s p o s e of R. - M i s f i t t i n g the o b s e r v a t i o n s e° i s r e a d i l y a c h i e v e d by winnowing, say, the K s m a l l e s t e i g e n v a l u e s w i t h t h e i r a s s o c i a t e d e i g e n v e c t o r s , t h a t i s , t r u n c a t i n g m a t r i c e s R, A and R T t o s i z e 74 (NxM), (MxM) and (MxN) r e s p e c t i v e l y , with M=N-K. The c o e f f i c i e n t s ac c o n s t r u c t e d from the truncated set of s p e c t r a l components are, a? = R ^ R j i 0 (A-2)' Using a c we can compute the c a l c u l a t e d o b s e r v a t i o n s , i . e . 1° = T a c (A-3) Secondly, c o n s i d e r the x 2 value d e f i n e d by X 2 = Z ( e T - e P ) 2 = ||A?|| 2 (A -4 ) with A? = e'-e° and E ( x 2 } - N f o r N>5. The l e n g t h of the vecto r Ae i s not changed upon r o t a t i o n . T herefore we can p r o j e c t e 1 and e° onto the e i g e n v e c t o r s of R, i . e . X 2 = ||R Tt\" - R T e ° | | 2 o r X 2 = £(e? - S ? ) 2 (A-5) F i n a l l y , from (A-3) and (A-2) we w r i t e , e 6 = R A R T R M A : 1 R j t ° P r e m u l t i p l y i n g t h i s equation by R T and w r i t i n g down the n o t a t i o n f o r r o t a t e d o b s e r v a t i o n s we o b t a i n , ec = AR TR wA;'e 0 , (A-6) A from which we r e a l i z e that ef = e° f o r i=1,...,M and ef = 0 f o r i=M+1,...,N. Hence (A-5) g i v e s , X 2 =z(i?)2 75 For complex data we use, X 2 = Ze^e?*\" (A-7) where * i n d i c a t e s the complex conjugate. S t a r t i n g with M=N-1 we form the summation given in equation ( A - 7 ) , and keep adding terms u n t i l t h i s summation y i e l d s the c l o s e s t value to N (number of o b s e r v a t i o n s ) . The f i n a l index M g i v e s the number of eigenvalues and e i g e n v e c t o r s to be r e t a i n e d in the c a l c u l a t i o n of a c . These c o e f f i c e n t s , when used i n the c o n s t r u c t i o n , y i e l d a model which s a t i s f i e s the o b s e r v a t i o n s in a manner c o n s i s t e n t with o b s e r v a t i o n a l e r r o r s . 7 6 APPENDIX B E f f e c t s of the Weighting Function Q and Standard D e v i a t i o n Values on Model C o n s t r u c t i o n A f t e r i n t r o d u c i n g the standard d e v i a t i o n values o- and weighting f u n c t i o n Q, our o r i g i n a l problem e,- = has been m o d i f i e d t o, e ; = (B-1) where, e; = e; / O ; G* = G ;Q/a; m+ = Q\"1m Q\"1 = 1/Q with Q=k;°- 5K 0 0- 5 (k r b) f o r the (forward) s m a l l e s t problem Q=K,°' 5(k rb) f o r the (forward) f l a t t e s t problem and b an a r b i t r a r y p o s i t i v e r e a l number. In t h i s appendix we h i g h l i g h t the e f f e c t s of Q with a given b v a l u e , and of a- on the c o n s t r u c t e d model m=U(cj,k r). 1. E f f e c t s of Q on the s i z e of m. The r o l e of the constant b i n a t t e n u a t i n g l a r g e h o r i z o n t a l wavenumber components i s p o r t r a y e d in F i g u r e B.1. T h i s f i g u r e shows the p l o t s of the m o d i f i e d B e s s e l f u n c t i o n s K 0 ( r ( 7 ) ) and 77 K , ( r ( 7 ) ) , where the argument i s d e f i n e d by r ( 7 ) = b w s i n ( 7 ) / V with 0 . 1 ° < 7 < 9 0 ° , cj=407rrad/s, V=1500 m/s , and b assumes the values 1 and 10. C l e a r d i f f e r e n c e s on a t t e n u a t i o n r a t e imposed by these f u n c t i o n s are observed, such that the s i z e of the c o n s t r u c t e d model w i l l be s i g n i f i c a n t l y a f f e c t e d . In what f o l l o w s we d i s c u s s the s m a l l e s t model problem, whose weighting f u n c t i o n i n v o l v e s K 0 . 0 . 0 9 0 . 0 0 . 0 9 0 . 0 A X I S I N D E G R E E S P X I S I N D E G R E E S ( a ) ( b ) FIGURE B.I M o d i f i e d B e s s e l f u n c t i o n s (a) K 0, and (b) K, f o r two d i f f e r e n t b values i n the argument r ( 7 ) (r( 7 )=bwsin( y ) / V , tj=407rrad/s and V=1500m/s). Large b values s e v e r e l y attenuate l a r g e h o r i z o n t a l wavenumbers and decrease the l i n e a r independence of the k e r n e l s . The m i n i m i z a t i o n of ||m*|| r e q u i r e s that (Backus, 1970), | |Q- 'm| |,kr) i s c o n f i n e d to those whose high wavenumbers are s t r o n g l y attenuated. T h i s model a t t e n u a t i o n i s more severe f o r weighting f u n c t i o n s Q with l a r g e b va l u e s . Homogeneous plane waves have h o r i z o n t a l wavenumbers r e s t r i c t e d between 0 and C J / V ( i . e . angles of emergence between 0° and 90°). For t y p i c a l e x p l o r a t i o n seismic work, C J / V i s smaller than u n i t y and hence, with small b va l u e s , these wave components are not s e v e r e l y attenuated by Q\"1 in the model m. On the other hand, inhomogeneous waves have wavenumbers between u / V and °° ( i . e . complex angles between 90° and 90°-i<»: Brekhovskikh, 1960). When k r>u ) / V but i s \"reasonably s m a l l \" , these waves are s t i l l c o n t r o l l e d by the data equation (B-1). However, as k r becomes l a r g e r the inhomogeneous waves are i n c r e a s i n g l y attenuated by the requirement s p e c i f i e d i n (B-2) so that t h e i r amplitudes w i l l decrease e x p o n e n t i a l l y . I t i s c l e a r that the \" t r a n s i t i o n \" value of kr at which c o n s t r a i n t (B-2) predominates (B-1) depends on the chosen b value, that i s to say | |k r°- 5(k rb)°- 2 5exp(k rb/2)U(cj,k r) | | has to be kept f i n i t e as k r becomes l a r g e (k r-*•»). The preceding d i s c u s s i o n a p p l i e s to the forward f l a t t e s t problem as w e l l . In t h i s case, the weighting f u n c t i o n Q a f f e c t s the d e r i v a t i v e of m, or e q u i v a l e n t l y , the rate of change of the 79 c o n t r i b u t i o n of the homogeneous and inhomogeneous plane waves at a given angular frequency GJ. 2. E f f e c t s of Q on the inner product matrix T. The elements of the inner product matrix T for the (forward) s m a l l e s t and f l a t t e s t problems are i n v e r s e l y p r o p o r t i o n a l to [ (r ; 2+b 2 + r-2) 2 - 4 r - 2 r J 2 ], where r ; and rj are the o f f s e t s corresponding to the i ' t h and j ' t h geophone l o c a t i o n s such that r- T2 2>. . .>ruiJ . For b<T 2 2>...>C„. T h i s behaviour of the d i a g o n a l elements of T provi d e s a n a t u r a l o r d e r i n g f o r the b a s i s f u n c t i o n s such that the long-wavelength s t r u c t u r e of the model i s c o n t r o l l e d by the b a s i s f u n c t i o n s a s s o c i a t e d with the l a r g e s t eigenvalues, whereas the f i n e s t r u c t u r e o r i g i n a t e s from those b a s i s f u n c t i o n s a s s o c i a t e d . with the sm a l l e s t e i g e n v a l u e s . D i v i d i n g each element of T by a- <7j the r e l a t i o n s , >T 2 2> . . . >rufJ do not n e c e s s a r i l y h o l d . In p a r t i c u l a r , f o r the case of small b value and l a r g e l y v a r y i n g standard d e v i a t i o n s , we do not expect that the eigenvalues of T arranged i n d e c r e a s i n g order w i l l correspond to i t s d i a g o n a l elements i n t h e i r o r i g i n a l order. The new order r e l a t i o n s w i l l depend on the r e l a t i o n s h i p s between the standard d e v i a t i o n s . For example, a very small o 2 value c o u l d make r v > J the l a r g e s t d i a g o n a l element and consequently, the f i r s t ordered eigenvalue w i l l e s s e n t i a l l y correspond to t h i s element. But the most important r e s u l t of t h i s r e o r d e r i n g of eig e n v a l u e s versus d i a g o n a l elements of T i s the consequent r e o r d e r i n g of the b a s i s f u n c t i o n s i>; . Hence, i n the above example, the l a s t weighted k e r n e l ( a s s o c i a t e d with YKtl) c o u l d become the f i r s t b a s i s f u n c t i o n and consequently small-wavelength model s t r u c t u r e w i l l stem from T h i s i s i l l u s t r a t e d i n F i g u r e B.5 where we have p l o t t e d on Panel (a) some b a s i s f u n c t i o n s f o r the case a,=1, whereas on Panel (b) we have d i s p l a y e d the same b a s i s f u n c t i o n s with O,=10% of the maximum (amplitude) s p e c t r a l value of the 84 i ' t h t r a c e of the data presented i n Fi g u r e 3. We summarize by s t a t i n g that in the process of winnowing b a s i s f u n c t i o n s , the standard d e v i a t i o n values w i l l play a major r o l e in determining the type of inf o r m a t i o n to be included in the c o n s t r u c t e d model. Thus a p p l i c a t i o n of a-, v a l u e s to a given PWD problem should be e x e r c i s e d with c a u t i o n . 85 Panel a 1 3 H E R 7 Z Panel b 0. 3 E ' 0 2 t _ j _ _ L _ i _ ( _ L i — i — L — i O . U E - O S t - 0 . 2 E - 0 3L_i_i—I I 1 1 — J — I — I — I - 0 . 5 E » 0 3 J — i — I — I — I — I — i — J — I — u 2 . 9 0 . 2 . 9 0 . p p f - . n i , , , , . i i i i i 0 . 2 E « 0 3 i — i — i — i — i — i 0 . 3 E - > 0 3 L _ i — j i i i i i i i r 0 . 2 E * 0 3 i ^ - i i — I — i — i — I — i — i — t 25 - 0 . 3 E • 0 3J1 I I I I I—I—1_ 2 . 0 . 2 E » 0 3 i i I i — i — i — i — i - 0 . 2 E - » 0 3 J 25 _i I - 0 . 3 E * 0 3 1 I 1 I I—I—I—I—1— 9 0 . 2 . 9 0 . 0 . U E * 0 3 i i I i i i i — i — I — L _ - 0 . 1 E * 0 3 J -9 0 . 2 . ( H O R I Z O N T A L R X E S I N D E G R E E S ! FIGURE B.5 Ba s i s f u n c t i o n s 1, 13, 25 and 35 corresponding to the forward smallest-model c o n s t r u c t i o n . Standard d e v i a t i o n s are set to (a) 1 f o r a l l input t r a c e s , and (b) percentages of the maximum s p e c t r a l amplitude of each t r a c e (see t e x t f o r d e t a i l s ) . N o t i c e that the r e - o r d e r i n g of the b a s i s f u n c t i o n s i s such that i n (a) the f i r s t b a s i s f u n c t i o n c o n t r i b u t e s with long-wavelength model s t r u c t u r e , whereas i n (b) t h i s b a s i s f u n c t i o n g i v e s short-wavelength s t r u c t u r e . 86 APPENDIX C Inner Product Matrix f o r the Forward Smallest Model C o n s t r u c t i o n The c o n s t r u c t i o n of the forward s m a l l e s t model r e q u i r e s the e v a l u a t i o n of the elements of the inner product matrix X. Using the k e r n e l s from equation (2-9b) we have, r:. = = [l/a,-o ;] f k r K 0 ( k r b ) J 0 ( k r r ; ) J 0 ( k T rj )dk r (C-1) 0 and from Gradshteyn and Ryzhik, 1980, equation 6.578.15 we obta i n , r;- = [ l / a ; a - ] / [ (r?+b 2 + r i 2) 2-4r i 2r j 2]°- 5 (C-2) 87 APPENDIX D The Forward F l a t t e s t Model: G l o b a l Development T h i s appendix d e s c r i b e s the s o l u t i o n s to the set of problems which are encountered in the c o n s t r u c t i o n of the f l a t t e s t - m o d e l forward transform. In p a r t i c u l a r , the f o l l o w i n g problems are undertaken: 1. Computation of the f l a t t e s t model k e r n e l s H;. 2. E v a l u a t i o n of the term H; m°f . 3. C a l c u l a t i o n of the inner product K r 4. S o l u t i o n to the i n d e f i n i t e i n t e g r a l JQ 2H;. 5. E v a l u a t i o n of the i n t e g r a t i o n constant o f equation (2-14). 1. Computation of the f l a t t e s t model k e r n e l s H ;. With the i d e n t i f i c a t i o n , = k r J 0 iK r r )/o: we e v a l u a t e the i n d e f i n i t e i n t e g r a l , H; = [ ] / k ' J 0 ( k ' r ; )dk' Upon change of v a r i a b l e r ' = k'r ; we get, H; = [ \\ / a . r , 2 ] / r ' J 0 ( r ' ) d r ' and from Gradshteyn and Ryzhik, 1980, equation 5.56.2 we o b t a i n , H ; = [ l / a ; r ; ] k r J , ( k r r ; ) (D-1) 88 2. E v a l u a t i o n of the term H ;mj . Using (D-1) and the asymptotic behaviour of J , ( k r r r ) f o r l a r g e argument, the f i r s t term of the r.h.s. of equation (2-11) fo r kr-»°° behaves as, U(oj,kr-°°) l i m {[ 1/ff, r,- ] k r [ 2 A k r r , ] ^ } = U(u,k r») l i m {2k r , , 2/ff, 7r1'2r;3/2} (D-2) T h i s e x p r e s s i o n goes to zero provided U(a>,kr—°°) goes to zero f a s t e r than k r \" 1 / 2 f o r a l l f r e q u e n c i e s (because the modified B e s s e l f u n c t i o n s K 0(x) and K,(x) are not d e f i n e d at x=0 we exclude both o>=0 and 7=0°). R e c a l l i n g that k r=wsin (y)/V, and c o n s i d e r i n g a b a n d - l i m i t e d source, kr-*°° represents i n f i n i t e l y -a t t e n u a t e d inhomogeneous waves. In the present work we set • H; mT = 0 Then we w i l l be concerned with models U(w,k r) which go to zero f a s t e r than k r~ 1' 2 as kr-»°°. 3. C a l c u l a t i o n of the inner product matrix. With Hj as given i n (D-1) and Q = K,°- 5(k rb), the e n t r i e s of the inner product matrix T are given by, r;- = = [ 1/r- r- o.o- ]|kr2K, ( k r b ) J , ( k r r . ) J , ( k r r j )dk r and from Gradshteyn and Ryzhik, 1980, equation 6.578.15 we have, ri-=[\\/oios] 4 b / [ ( r 2 + b 2 + r . 2 ) 2 - 4 r 2 r 2 ] 1 - 5 (D-3) 89 K r 4. E v a l u a t i o n of the i n d e f i n i t e i n t e g r a l JQ 2H-. The s o l u t i o n to the i n d e f i n i t e i n t e g r a l appearing i n the l . h . s . of equation (2-15) can be d e r i v e d from the f o l l o w i n g ( i n d e f i n i t e ) i n t e g r a l given i n Gradshteyn and Ryzhik, 1980, equation 5.54.1, J y Z p ( a y ) B P ( c y ) d y = {cyZ ?(ay )B*_, (cy) - ay Z?_, (ay ) B P (cy)}/[ a 2 - c 2 ] (D-4) where Zv and B ? are any B e s s e l f u n c t i o n of order p, and a and c are c o n s t a n t s . For p=1 we i d e n t i f y , Z, = J , B, = H,\"' H , ( 1 ) i s the f i r s t Hankel f u n c t i o n of order one. (D-4) now reads as, J y J , ( a y ) H , ' 1 ' ( c y ) d y = {cyJ,(ay)H 0< 1 ' ( c y ) - a y J 0 ( a y ) H , ( 1 ' ( c y ) } / [ a 2 - c 2 ] (D-5) but from Gradshteyn and Ryzhik, 1980, equations 8.407.1 and 8.407.2 we have, H 0 ( 1 ) ( i b y ) = -[2/TT] i K 0 ( b y ) (D-6a) H , ( 1 » ( i b y ) = -[2/TT] K,(by) • (D-6b) Using (D-6) and i d e n t i f y i n g c=ib, (D-5) transforms to, -[2/TT] J y J , (ay)K, (by)dy = { ( i b ) y J , ( a y ) [ - 2 / i r i K 0 ( b y ) ] - ayJ 0(ay)[-2/w K,(by)]} / [ a 2 - ( i b ) 2 ] 90 that i s to say, f y j , ( a y ) K , ( b y ) d y = - {byJ,(ay)K 0(by) + a y J 0 ( a y ) K , ( b y ) } / [ a 2 + b 2 ] (D-7) from which a f t e r making y = k r and a = r-( we o b t a i n the r.h.s. of equation (2-15). 5. The i n t e g r a t i o n constant of equation (2-14). We show that with the c o n d i t i o n U(u, kr-»°°) = 0 the constant of i n t e g r a t i o n C i s equal to zero. From equations (2-14), (D-1) and (D-7) (with y = k r and a = r,- ) we see t h a t , C = -Z[0,-/a ;rj ] [ l / ( r ? + b 2 ) ] { 1 im{ bk r J , ( r r k r ) K 0 (bk r )} + l i m { r ; k r J 0 ( r r k r)K, (bk r)} } +U(co,k r^») (D-8) Asymptotic expansions f o r l a r g e arguments of the i n v o l v e d B e s s e l f u n c t i o n s and the m o d i f i e d B e s s e l f u n c t i o n s read as (Abramowitz and Stegun, 1970), J 0 ( r ( . k r ) ~ [2/7rr ; k r ] 0 - 5 {cos(r,.k r) + 0 ( | r - k r | - 1 )} J , ( r ; k r) ~ [2/jrr f k r ] °-5 { s i n ( r ; kr ) + 0( | r. kr | \" 1 ) } K 0 ( b k r ) ~ [7r/2bk r] 0- 5 e x p ( - b k r ) { l + 0 ( | b k r | - 1 ) } K,(bk r) ~ [7r/2bk r]°- 5 e x p ( - b k r ) { l + 0 ( | b k r | - 1 ) } T h e r e f o r e , the l e a d i n g behaviours of the l i m i t i n g e x p r e s s i o n s i n (D-8) are given by, l i m {bk r [2/irv, k r ] °-5 [ 7r/2bk r ] °-5 exp(-bk r)} + l i m {r f k r [ 2 / 7 r r ; k r ] ° - 5 [ 7 r / 2 b k r ] 0 - 5 exp(-bk r)} = 0 (D-9) that i s to say, C = 0 91 APPENDIX E Inner Product M a t r i c e s f o r the I n v e r s e S m a l l e s t and F l a t t e s t Model C o n s t r u c t i o n s The c o m p u t a t i o n of the i n n e r p r o duct m a t r i c e s f o r the i n v e r s e s m a l l e s t and f l a t t e s t problems p a r a l l e l s t h a t made f o r the f o r w a r d problems. Indeed, by r e p l a c i n g r ; by k r ; as the parameter and kr by r as the independent v a r i a b l e a p p e a r i n g i n the f o r m u l a t i o n of the f o r w a r d problems, one r e a d i l y o b t a i n s the i n n e r p r o d u c t m a t r i c e s f o r the i n v e r s e problems. In p a r t i c u l a r , the e n t r i e s of the T m a t r i x f o r the s m a l l e s t i n v e r s e t r a n s f o r m are g i v e n by, rVj(ej) = [Wo; 0 j ] / [ ( k 2 + b 2 + k 2 ) 2 - 4k r 2 k 2 . ] 0 - 5 (E-1) A f t e r the s u b s t i t u t i o n s k r (=cjsin (7,- ) / V and b=ojpc =a>sin (c ) / V , where c i s a c o n s t a n t such t h a t s i n ( c ) > 0 , and V i s the s u r f a c e P-wave v e l o c i t y , (E-1) y i e l d s r ; j(cj) = V 2 / C J 2 {[ I/a; c j j ] / [ ( s i n 2 (7,. ) + s i n 2 ( c ) + s i n 2 ( 7 j ) 2 -4 s i n 2 ( 7 ; ) s i n 2 ( 7 i ) ] 0 - 5 } . We w r i t e rr(u) = V 2 / G J 2 F • (E -2) where, r r = [ 1/of a; ]/[ ( s i n 2 (7, ) + s i n 2 ( c ) + s i n 2 (7. ) 2 -4 s i n 2 (7- )-sin 2 (75 ) ] 0 - 5 92 Since the frequency-dependent term i n (E-2) i s f a c t o r e d out of the inner product matrix, the inverse r~ 1 (a>) = C J 2 / V 2 T\" 1 i s c a l c u l a t e d with minimal computational e f f o r t . S i m i l a r l y , the e n t r i e s of the inner product matrix for the in v e r s e f l a t t e s t problem are given by, r,.(«) = [ l/o ; o: ]4b/[ ( k 2 + b 2 + k 2 ) 2 - 4 k r 2 k 2 ] 1 - 5 (E-3) Upon the s u b s t i t u t i o n s b = c j s i n ( c ) / V and kr- =u>sin(7r ) / V we o b t a i n from ( E - 3 ) , rv(cj) = V 5 / u 5 r ; j where T ; j = [ i/o; O j ] 4 s i n ( c ) / [ ( s i n 2 (7,- ) + s i n 2 ( c ) + s i n 2 (y± ) -4 s i n 2 (7 r ) s i n 2 (7^ ) ] '-5. "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0052984"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Geophysics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Plane-wave decomposition and reconstruction of spherical-wave seismograms as a linear inverse problem"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/24556"@en .