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Sparseness-constrained seismic deconvolution with curvelets Hennenfent, Gilles; Herrmann, Felix J.; Neelamani, Ramesh 2005

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Sparseness-constrained seismic deconvolution with Curvelets G. Hennenfent1, F. Herrmann1 and R. Neelamani2 1  EOS, University of British Columbia, Canada, 2 ExxonMobil Upstream Research Company, Houston, TX, USA  Summary Con t i nu i t y a l ong re f lec to rs in se i sm i c images is used v ia Cu r ve l e t r ep r esen t a t i on t o s t ab i l i z e t h e convo l u t i o n ope ra t o r i n v e r s i on . The Cu r ve l e t t r ans f o rm i s a new mu l t i s ca l e t r ans f o rm t h a t p rov i des spa r se r ep r esen t a t i ons f o r images t ha t comp r i s e smoo th ob j e c t s sepa ra t ed by p i e ce - w i se smoo t h d i s con t i nu i t i e s ( e . g . se i sm i c images ) . Our i t e r a t i v e Cur ve l e t - r egu l a r i z ed deconvo l u t i o n a l go r i t hm comb i nes con j uga t e g rad i en t - based i n ve r s i on w i t h no i se r egu l a r i z a t i o n pe r f o rmed us i ng non - l i n ea r Cu rve l e t coe f f i c i en t t h r e sho l d i ng . The t h r e sho l d i ng ope ra t i on enhances t h e spa r s i t y o f Cur ve l e t r ep r esen t a t i ons . We show on a syn t he t i c examp l e t ha t ou r a l go r i t hm p rov i des imp r oved r e so l u t i o n and con t i n u i t y a l ong r e f l e c t o r s as we l l as r educed r i n g i ng e f f e c t compa red t o t h e i t e r a t i v e Wiene r - based deconvo l u t i on app roach . In t r oduct i on I n t h i s pape r , we add ress t h e c l a s s i c a l The f o rwa rd p rob l em i s  d i s c r e t e - t ime deconvo l u t i on p rob l em .  (1)  d = Km + n  whe re d i s t h e da t a , K t he convo l u t i on ope ra t o r ( c y c l i c ma t r i x ) , m t h e se i sm i c r e f l e c t i v i t y and n ze ro - mean add i t i v e wh i t e Gauss i an no i se w i t h va r i ance σ 2. The assump t i on o f wh i t e no i se can be r e l a xed t o co l o r ed no i se as l o ng as i t s cova r i ance ma t r i x i s nea r d i agona l i n cu r ve l e t f r ames . The i n ve r se p rob l em i s t o de t e rm i ne m g i ven d and K . No t i c e t h a t we do no t cons i de r t h e p rob l ems o f b l i nd deconvo l u t i on and sou r ce s i gna t u r e es t ima t i on . A na i ve app r oach t o i n ve r se K - 1 as  ob t a i n  an es t ima t e  fo r  m÷ wou l d be us i ng t h e ope ra t o r  m÷ = K - 1d = m + K - 1n  +  (2)  o r K , t h e pseudo - i n ve r se , f o r a non - i n ve r t i b l e K . Un fo r t una t e l y , t h e d i s c r e t e deconvo l u t i o n p r ob l em i s i l l - cond i t i o ned . I n t h a t case , t he va r i ance o f K - 1n i s ÷ an unsa t i s f a c t o r y deconvo l u t i on es t ima t e . l a r ge mak i ng m S ince t he 60 ’ s , deconvo l u t i o n f i l t e r s based on t h e Wiene r t h eo r y a re w ide l y used . Wiene r f i l t e r s min im i ze t h e mean - squa red - e r r o r (MSE) [ 15 ] . As ea r l y as t h e mid - 70 ’ s t o ea r l y 80 ’ s , t h i s min ima l ene rgy app roach was gene ra l i z ed t owa rds an l 1- no rm min im i za t i on , des i gned t o b r i ng t h e sp i k yness o f t h e r e f l e c t i v i t y [ 3 , 10 , 11 ] . I n t he image p rocess i ng and wave l e t commun i t i e s , s im i l a r i d eas eme rged i n t h e ea r l y 90 ’ s whe re deconvo l u t i on p rob l ems a re so l ved by a l so exp l o i t i n g spa r seness o f t he mode l bu t now on ce r t a i n bases t ha t a re des i gned t o be spa r se on t h e mode l ( s ee e . g . [ 9 , 14 , 16 ] ) . Our app roach bo r r ows f r om bo t h t h e min ima l s t r u c t u r e concep t s i n t r oduced i n geophys i c s and f r om above r e cen t i d eas i n image p rocess i ng , whe re t h e spa r seness o f ce r t a i n f r ame expans i ons i s used t o r egu l a r i z e deconvo l u t i on [ 4 , 14 ] . S ince t h e Ea r t h can be cons i de r ed t o cons i s t o f r e f l e c t o r s on p i e ce w i se smoo t h cu r ves , r e cen t l y deve l oped cu r ve l e t f r ames a re t he app rop r i a t e cho i ce . We beg i n by p rov i d i ng a b r i e f ove r v i ew o f cu r ve l e t f r ames . We t h en p resen t ou r cu r ve l e t - r egu l a r i z ed deconvo l u t i on a l go r i t hm and d i s cuss i ts p rac t i c a l imp l emen t a t i on . An i l l u s t r a t i v e syn t he t i c examp l e f o l l ows . Curvele t  f rames  The cu r ve l e t t r ans f o rm i s a new membe r i n t h e f am i l y o f Compu ta t i ona l Ha rmon i c Ana l y s i s t o o l s [ 1 , 2 ] . T i gh t cu r ve l e t f r ames we re i n i t i a l l y des i gned t o p rov i de op t ima l l y spa r se r ep r esen t a t i ons f o r ob j ec t s t ha t a re smoo t h excep t a l ong Evolving Geophysics Through Innovation  1  smoo t h cu r ve - l i k e d i s con t i nu i t i e s ( e . g . image w i t h edges ) . Cu rve l e t s a re ob t a i ned by pa r t i t i o n i ng t h e 2D Fou r i e r p l ane i n t o dyad i c co r onae and sub pa r t i t i o n i ng t hose i n t o angu l a r wedges . Cu rve l e t s obeys a pa rabo l i c sca l i n g l aw – a t sca l e 2- j , each e l emen t has an e f f e c t i v e suppo r t o f l e ng t h 2- j / 2 and w id t h 2- j . The r e su l t i n g cu r ve l e t f r ames a re mu l t i - sca l e , mu l t i - d i r e c t i o na l , h i gh l y an i so t r op i c , and l o ca l i z ed bo t h i n space and f r equency . Cu r ve l e t s can pe r f o rm decompos i t i o n and r e cons t r u c t i on o f any f u nc t i on much l i k e an expans i on i n an o r t hono rma l bas i s , bu t i n con t r a s t , cu r ve l e t r ep r esen t a t i ons a re mode ra t e l y r edundan t . On t h e p rac t i c a l s i de , t h e r e ex i s t s f a s t O(Nl o g N) a l go r i t hms t ha t a l l ow f o r a decompos i t i o n o f n- by - m images f w i t h n* m=N. These a l go r i t hms a re nume r i c a l l y t i g h t and have an exp l i c i t cons t r u c t i o n f o r t h e ad j o i n t t h a t equa l s t h e pseudo i n ve r se  C *×= C + × and C *C = I  whe re C r ep r esen t s t h e cu r ve l e t t r ans f o rm .  Evolving Geophysics Through Innovation  (3)  2  Figu re 1: Five curve l e t s  at d i f f e r e n t  sca l e s and or i e n t a t i o n  Cons i de r a f unc t i on f o f two va r i ab l e s t h a t i s p i e ce - w i se tw i c e d i f f e r en t i ab l e . Fu r t he r , l e t t h e d i s con t i nu i t y cu r ves sepa ra t i ng t h e smoo t h p i e ces o f f a l so be p i e ce - w i se tw i ce d i f f e r en t i ab l e . F rom t h e non - l i n ea r pe r spec t i v e , t he op t ima l app r ox ima t i on r a t e equa l s [ 5 ] 2  f - f ko µ k - 2 for k ® ¥ (4) 2 i s t h e pa r t i a l r e cons t r u c t i on o f f us i ng t h e k l a r ges t t e rms i n t he  whe re f ko bas i s . Cu r ve l e t f r ames ach i e ve [ 2 ]  f - f kc  2 2  3  £ Ck - 2 (log k ) for k ® ¥  (5)  I n wo rds , we quo t e f r om [ 2 ] “ t he r e i s no bas i s i n wh i ch coe f f i c i en t s o f an ob j e c t w i t h an a rb i t r a r y C 2 s i ngu l a r i t y wou l d decay f a s t e r t h an i n a cu r ve l e t f r ame ” . Fo r compa r i son , t h e decay r a t e f o r Fou r i e r coe f f i c i en t s i s O( k - 1 / 2 ) and f o r wave l e t coe f f i c i en t s O( k - 1 ) . Curvelet shrinkage-based signal estimation  Cons i de r f r amewo r k w i t h K = I ( i . e . deno i s i ng p rob l em ) . I n a bas i s o f wave l e t s , a s imp l e sh r i n kage ( i . e . so f t - t h r e sho l d i ng ) o f t he wave l e t coe f f i c i e n t s o f t he no i s y da t a min im i zes a quad ra t i c d i s t ance t o t h e da t a pena l i z ed by a l 1 - no rm [ 8 ] . The va r i a t i o na l p rob l em i s  m÷ = argmin m  1 2 d - m 2 + l Wm 1 2  (6 )  whe re r ep r esen t s t h e wave l e t t r ans f o rm . In cu r ve l e t f r ames , so f t W t h r e sho l d i ng so l ves on l y ( 6 ) app rox ima te l y bu t a l r eady p rov i des a good es t ima t e f o r t h e o r i g i na l s i gna l [ 1 3 ] . Curvele t s for seismic deconvolu t i on Ma in f ea t u r e s i n se i sm i c images co r r e spond t o uncon f o rm i t i e s i n t h e Ea r t h ’ s subsu r f a ce wh i ch t e nd t o f o l l ow p i ece - w i se smoo t h cu r ves , a l l ow i ng f o r p i n ch ou t s and f a u l t s . Hence t he cu r ve l e t r ep r esen t a t i on o f se i sm i c images can be expec t ed t o be spa r se . We use t h i s add i t i o na l p i ece o f i n f o rma t i on t o f i l l t he nu l l space o f t h e ope ra t o r and make t h e i n ve r s i on more s t ab l e . The f o rwa rd p rob l em f r amewo r k i s r e f o rmu l a t ed as (7 )  d = Fx + n whe re x i s a vec t o r o f cu r ve l e t coe f f i c i en t s such t ha t The va r i a t i o na l p rob l em now becomes [ 4 ]  x÷ = argmin x  1 2 d - Fx 2 + l x 1 2  m = C* x ,  and  F×= KC ×. (8 )  I n wo rds , we wan t t o deconvo l ve t he da t a and s t ab i l i z e t h e p rocess by impos i ng a spa r s i t y cons t r a i n t on t h e cu r ve l e t r ep r e sen t a t i on o f t h e mode l . I n t u i t i v e l y speak i ng , ( 8 ) so l ves t h e deconvo l u t i on p rob l em ” c u r ve l e t - w i se ” bu t t he exac t so l u t i on i s no t t r i v i a l due t o t he non - d i f f e r en t i ab i l i t y o f t he l 1- no rm a t t he o r i g i n . A popu l a r cho i ce i s t h e I t e r a t i v e Rewe i gh t ed Leas t - Squa res ( IRLS ) Evolving Geophysics Through Innovation  3  me thod t ha t i s known t o g i ve a good app rox ima t i on t o t h e l 1 - no rm [ 6 , 12 ] t o so l v e ( 8 ) . We p ropose a me thod i n sp i r ed by t h e wor k i n [ 4 ] . We comb i ne t he Con j uga t e Grad i en t (CG ) me thod w i t h so f t t h r e sho l d i ng t o ob t a i n an app rox ima t e so l u t i o n (8 ) . F i r s t , t h e CG-me thod i s app l i ed t o t he no rma l equa t i ons F * d = F * Fx + F * n . Typ i ca l l y , t h e no i se t h a t ge t s amp l i f i e d by t h e sma l l s i ngu l a r va l ues o f F * F i s r egu l a r i z ed by s t opp i ng CG p rema tu r e l y ; t h e s t opp i ng c r i t e r i on i s p ropo r t i ona l t o t h e no i se l e ve l . Our app roach r egu l a r i z es t he no i se by so f t t h r e sho l d i ng t h e cu r ve l e t coe f f i c i en t s a f t e r a l im i t ed numbe r o f CG i t e r a t i ons and t h en r e s t a r t i ng CG w i t h t h e t h r e sho l ded es t ima t e . Bes i des r emov i ng t h e no i se , so f t t h r e sho l d i ng enhances t h e sp i k yness (m i n im i zes t h e l 1- no rm o f t he coe f f i c i en t s ) . Th i s i t e r a t i v e CG r egu l a r i z a t i on app rox ima t e l y so l ves ( 8 ) wh i l e p rese r v i ng t h e f r equency i n f o rma t i on assoc i a t ed w i t h t h e sma l l s i ngu l a r va l ues . The t h r e sho l d l e ve l i s t a ken l a r ge a t t h e beg i nn i ng and made sma l l e r t owa rds t h e end .  Evolving Geophysics Through Innovation  4  The a l go r i t hm i s g i ven be l ow : 1. I n i t i a l i z e t h r e sho l d Γ0 , f i r s t guess x 0 , se t be tween t h r e sho l d i ngs J and numbe r o f i t e r a t i o ns Lmax 2.  numbe r  of  CG i t e r a t i on s  Fo r i = 1: Lmax •  x i = CG( x÷i- 1,J) ,  •  So f t - t h r e sho l d t h e coe f f i c i en t s  x÷i .  x i w i t h t h e t h r e sho l d Γi and ob t a i n  The r e f l e c t i v i t y es t ima t e i s g i ven by t h e i n ve r se t r ans f o rm  ÷ = C* x ÷. m  Resul ts The no i s y da t a a re ob t a i ned by convo l v i ng t h e R i c ke r wave l e t w i t h t h e Marmous i se i sm i c r e f l e c t i v i t y and add i ng wh i t e Gauss i an no i se ( SNR~6dB) . As a r e f e r ence , t h e Wiene r f i l t e r i s  Hö(w) =  Gö(w) 2 P (w) , Gö(w) + n Pm (w)  (9 )  whe re G(ω ) i s t h e Fou r i e r t r ans f o rm o f t h e se i sm i c sou r ce s i gna t u r e , Pm( ω ) t he powe r spec t r um o f t h e mode l , and Pn ( ω ) t h e powe r spec t r um o f t h e no i se . I n p rac t i c e , Pm( ω ) i s unknown so we i t e r a t i v e l y es t ima t ed i t us i ng t h e me thod desc r i bed i n [ 7 ] . F i gs . 2c & 2d show t h e i t e r a t i v e Wiene r - based deconvo l u t i on es t ima t e and F i gs . 2e & 2 f t he es t ima t e ob t a i ned us i ng t he p roposed a l go r i t hm . Un l i k e t he Wiene r f i l t e r , ou r me thod i s ab l e t o r e cons t r u c t f r equency componen t s wh i ch have been deg raded by t h e no i se by exp l o i t i n g t h e spa r s i t y o f t h e r e f l e c t i v i t y ’ s cu r ve l e t r ep r esen t a t i on . As a r e su l t , t he f r e quency con t en t o f t h e deconvo l u t i on es t ima t e i s imp roved as we l l as t he con t i nu i t y a l ong r e f l e c t o r s . I n pa r t i c u l a r , we po i n t ou t t h a t t he t a r ge t a rea i s be t t e r r e so l v ed ( F i g s . 2d & 2 f ) . Moreove r , t h e r i ng i ng e f f e c t i s a l so r educed bu t some a r t i f a c t s , r e l a t ed t o t h e r e s i dua l no i se and Gibbs - l i k e e f f e c t s , a re s t i l l p resen t i n t he cu r ve l e t deconvo l ved image . Conclus ions We deve l oped and demons t r a t ed a new i t e ra t i ve cu r ve l e t - r egu l a r i z ed deconvo l u t i o n a l go r i t hm t ha t exp l o i t s con t i nu i t y a l ong r e f l e c t o r s i n se i sm i c images . The mo t i va t i on f o r t he cu r ve l e t app roach s t ems f r om t h e r ea l i z a t i on t h a t c l a s s i c a l me thods l i k e Wiene r f i l t e r i ng a re no t ab l e t o p rese r ve f r equency con t en t s deg raded by no i se . Spa r seness i n t h e cu r ve l e t doma i n , on t h e o t he r hand , b r i ngs ou t t h e s i gna l componen t s mos t l y i n f l u enced by t h e no i s e . Our a l go r i t hm can be app l i ed t o a w ide r ange o f app l i c a t i ons as l o ng as 1 ) t he mode l i s known t o be o t he rw i se smoo t h hav i ng d i s con t i n u i t i e s a l ong p i e ce - w i se smoo t h cu r ves and 2 ) t h e cova r i ance ma t r i x o f t h e no i s e i s nea r d i agona l i n cu r ve l e t f r ames . So f a r , ou r r egu l a r i z a t i o n f o cused on enhanc i ng t h e spa r seness o f t h e cu r ve l e t coe f f i c i en t s . The nex t s t ep o f ou r r e sea r ch w i l l be t o exp l o i t appa ren t con t i nu i t y a l ong r e f l e c t o r s wh i ch , based on ou r expe r i ence w i t h con t i nu i t y - enhanced imag i ng , w i l l f u r t he r r educe t h e p rev i ous l y men t i oned ar t i f ac ts . Acknowledgements The au t ho r s wou l d l i k e t o t h ank t h e au t ho r s o f t h e D ig i t a l Cur ve l e t T rans f o rm (Candes , Donoho , Demane t and Y ing ) . Th i s wor k was ca r r i ed ou t as pa r t o f t he S INBAD p ro j e c t w i t h f i n anc i a l suppo r t , secu r ed t h r ough I TF ( t he I ndus t r y Techno l ogy Fac i l i t a t o r ) , f r om t h e f o l l ow i ng o rgan i za t i ons : BG Group , BP , ExxonMob i l , SHELL . Add i t i o na l f u nd i ng came f r om t h e NSERC D i s cove r y Gran t s 22R81254 . References [1 ] E. Candes and D. Donoho . Curve l e t s : A surp r i s i n g l y ef f e c t i v e nonadap t i v e rep re sen t a t i o n of ob je c t s wi t h edges . Curves and Sur fa ces , 1999 . L. L. Schumaker et al . (eds ) , Vanderb i l t Unive r s i t y Press , Nashv i l l e , TN. [2 ] E. Candes and D. Donoho . New t i g h t f r ames of curve l e t s and opt ima l rep resen t a t i o n s of ob je c t s wi t h C2 s i ngu l a r i t i e s . Techn i ca l repo r t , Cal t e ch , 2002. [3 ] J . Clae r bou t and F. Mui r . Robus t model i n g wi t h er r a t i c data . Geophys i c s , 38(05 ) : 8 26–844 , 1973 . 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ForWaRD: Four i e r - Wavele t Regula r i z e d Deconvo l u t i o n fo r I l l - Cond i t i o n e d Sys tems . IEEE Transac t i o n s on Signa l Process i n g , 52(2 ) , Februa r y 2004. D. W. Oldenbu rg , S. Levy , and K. P. Whi t t a l l . Wavele t est ima t i o n and deconvo l u t i o n . Geophys i c s , 46(11 ) : 1 5 28–1542 , 1981 . M. D. Sacch i , D. R. Vel i s , and A. H. Cominguez . Min imum ent r opy deconvo l u t i o n wi t h f r e quency - domain cons t r a i n t s . Geophys i c s , 59(06 ) : 9 38–945 , 1994. J . Sca le s and A. Gersz t e nko r n . Robus t methods i n i nve r se theo r y . Inve r se prob l ems , 4:1071–1091 , 1988 . J . Sta r c k , E. Candes , and D. Donoho. The curve l e t t r ans f o rm fo r image deno i s i n g . IEEE Transac t i o n s on Image Process i n g , 11:670–684 , 2000 . J . - L . Sta r c k , M. Nguyen , and F. Murtagh . Wavele t s and curve l e t s fo r image deconvo l u t i o n : a combined approach . Signa l Process i n g , 83(10 ) : 2 279–2283 , 2003 . N. Wiener . The ex t r a po l a t i o n , i n t e r p o l a t i o n and smooth i ng of s ta t i o n a r y t ime ser i e s , wi t h eng i nee r i n g app l i c a t i o n s . Wile y , 1949. F. J . Her rmann . Seism i c deconvo l u t i o n by atomic decompos i t i o n : a paramet r i c approach wi t h spa r seness cons t r a i n t s . In t e g r . Compute r - Aided Eng. , 2003  Evolving Geophysics Through Innovation  6  Figu r e 2: Synthe t i c example of spar seness - cons t r a i n e d se i sm i c deconvo l u t i o n wi t h Curve l e t s and compar i s on wi t h i t e r a t i v e Wiener - based deconvo l u t i o n approach . (a ) Noisy data ( σ no i se =0.5 σ data i . e . SNR~6dB) . (b ) Close up on the rese r v o i r reg i o n i n the no i s y data . (c ) I t e r a t i v e Wiener - based deconvo l u t i o n es t ima t e (5 i t e r a t i o n s wi t h a regu l a r i z a t i o n paramete r of 10 - 4 to s tab i l i z e the power spec t r um es t ima t e at f r e quenc i e s where the convo l u t i o n opera t o r response goes to zero ) (d ) Close - up on the rese r v o i r reg i on i n the i t e r a t i v e Wiener - based deconvo l u t i o n es t ima t e . (e ) Sparseness - cons t r a i n e d deconvo l u t i o n es t ima t e wi th Curve l e t s (5 CG i t e r a t i o n s between th r e sho l d i n g s – 100 t imes ) ( f ) Close - up on the rese r v o i r reg i on i n the spar seness cons t r a i n e d deconvo l u t i o n es t ima t e wi t h Curve l e t s .  Evolving Geophysics Through Innovation  7  

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