Stable block Toeplitz matrix for the processing of multichannel seismic data

Transcription

1 Indian Journal of Marine Sciences Vol. 33(3), Sepember 2004, pp Sable block Toepliz marix for he processing of mulichannel seismic daa Kiri Srivasava* & V P Dimri Naional Geophysical Research Insiue, Uppal Road, Hyderabad , India *[ kiri_ngri@rediffmail.com] Received 29 Sepember 2003; revised 14 June 2004 Compuaion of deconvoluion operaors in he case of single channel secioned inpu/mulichannel seismic daa involves he inversion of a block Toepliz marix. The inversion of such a marix poses several problems. I is well esablished ha he error energy which measures he well posedness of he marix is seen o decrease wih an increase in he filer lengh. However, wih an increase in filer lengh he condiion number of he associaed marix increases. This means ha here is a rade off beween ill posedness and accuracy. The ill-posed problem has been made well posed by a process of (1) normalizaion of he block Toepliz marix and (2) by adding prewhiening parameer. The prewhiening parameer is aken as a few per cen of arihmeic or he geomeric mean of he main diagonal of he block Toepliz marix. Applicaion o a synheic as well as field seismic daa shows ha he condiion number of he associaed block Toepliz marix is reduced by a process of normalizaion and adding prewhiening parameer. Furher i is observed ha he condiion number is smaller when he prewhiening parameer is aken as a few per cenage of geomeric mean as compared o he arihmeic mean. Sabilizing he marix following he above procedure will help in obaining sable as well as accurae deconvoluion operaors. [Key words: Toepliz marix, mulichannel seismic, deconvoluion, filer lengh] Inroducion Mulichannel daa are acquired in many fields e.g. reflecion seismic, seismology, asronomy, sonar and radiomery. Various mulichannel filers have been designed for he suppression of coheren noise, sacking problems and velociy analysis. One algorihm used for he processing of mulichannel daa is known as he LWR (Levinson-Wiener- Robinson) algorihm which is fas and calculaes he required predicion filer for a ime-invarian sequence 1. However, for he ime-varying inpu sequence alernaive mehods o he LWR algorihm such as he gradien mehods have been proposed 2,3. The design of leas squares deconvoluion operaors for mulichannel seismic daa involves he inversion of he block Toepliz marix. The compuaion of he combined deconvoluion operaors for he imevarying sequence being secioned ino piece wise saionary secions also involves he block Toepliz marix 3,5. This block Toepliz marix is srucurally similar o he single channel auocorrelaion marix bu wih a difference ha is elemens are marices raher han scalars and he main propery is ha all he *Corresponding Auhor sub marices are idenical 6. In his invesigaion, a deailed sudy on he general properies of he block Toepliz marix regarding he sabiliy of he soluion and accuracy of he esimaed parameers, hrough he analysis of he condiion number has been carried ou. I has been well esablished ha when he condiion number is nearly one, he sysem is said o be well condiioned i.e. sable. However, unsable sysems are associaed wih large condiion number and are ermed ill condiioned. Hence, i becomes necessary o invesigae he behaviour of condiion number of a marix wih respec o various facors such as normalizaion of he block Toepliz marix and adding prewhiening parameer. The sabiliy of he block Toepliz marix has been discussed in ligh of he prewhiening parameer 7,8. Maerials and Mehods The compuaion of he deconvoluion operaors for a single channel secioned inpu and muli channel daa are designed using normal equaions ha show up he propery of he block Toepliz marix 2. The block Toepliz marix is mahemaically expressed as R P(0) P(1) = P ( 1) P (0) (1)

2 216 INDIAN J. MAR. SCI., VOL. 33, No.3, SEPTEMBER 2004 where P(0) = and P(1) = Rii (0) R (0) R (0) R (0) Rii (1) R (1) R (1) R (1) sonic daa and he covariance marix obained was normalized by: 1/ m ' m 1 τ = τ τ ii τ j= 1 /2 1/2 ( ) R ( ) R ( ) R ( )/( R ( ) R ( τ )) (3) The elemens of he marix are correlaion funcions where i/j indicae he number of channels or secions. Suppose we have wo secions/channels say X 1 () and X 2 (), hen he correlaion funcion is expressed as: R 11 (τ)=e[x 1 (+τ) X 1 ()] auo-correlaion funcion of he X 1 () R 12 (τ)=e X 1 (+τ) X 2 ()] cross-correlaion funcion of X 1 () and X 2 () R 21 (τ)=e[x 2 (+τ) X 1 ()] cross-correlaion funcion of X 2 () and X 1 () R 22 (τ)=e[x 2 (+τ) X 2 ()] auo-correlaion funcion of X 2 () where τ = lag, i.e τ = 0, 1, 2, 3. When τ = 0 and 1 we ge a wo-lengh filer which has a block-toepliz marix represened by Eq. (1) and each elemen of he marix R is a sub marix. In he design of accurae and sable deconvoluion operaors mahemaical operaions need o be performed on he associaed block Toepliz marix, R. The approach aken in his paper is o quanify he sabiliy of he block Toepliz marix hrough he analysis of is condiion number. The condiion number, CN, measures he well posedness of he sysem and is he raio of he maximum eigen value o he minimum eigen value 9. max CN = λ (2) λ min The sysem is said o be well condiioned if he condiion number is nearly one. However, in mos seismic deconvoluion sudies he block Toepliz marix associaed wih is design are associaed wih large condiion number. Hence, he behaviour of he condiion number is invesigaed wih respec o (1) normalizaion of he block Toepliz marix, (2) prewhiening parameer, and (3) lengh of he filer. Normalizaion of he block Toepliz marix In his sudy he esimaed covariance marix i.e. he block Toepliz marix is normalized 10, where he maximum likelihood mehod was applied o several where Rii () τ is he auocorrelaion of he i h channel and R () τ is he cross correlaion beween i h channel 1/ m and j h m channel and R ( τ ) is he geomeric j= 1 mean of he diagonal elemens of he block Toepliz marix. This normalizaion procedure forces he esimaed covariance marix o have equal diagonal elemens, a propery of he Toepliz marix. The prewhiening parameer The prewhiening parameer is applied o avoid he singulariy arising from he zero eigen value. In oher words, we add a small amoun of whie noise o sabilize he covariance marix i.e. prewhiening parameer is added o he main diagonal o make he inverse scheme sable. Prewhiening he Wiener filer improves he oupu signal o ambien noise raio, bu a he same ime i reduces he resoluion 11. The selecion of appropriae prewhiening parameer is a very edious procedure as seen in mos mehods like he ridge regression mehod 12, he singular value decomposiion 13,14 and he maximum likelihood mehod 15. In rouine seismic processing he prewhiening parameer is chosen as a few per cen of he arihmeic mean of he main diagonal. As he geomeric mean is known o resore he overall power of he covariance marix, he prewhiening parameer is aken as a few per cen of he geomeric mean of he main diagonal. Adding prewhiening parameer o he normalized block Toepliz marix resuls in '' ' R ( τ ) = R ( τ) + θi (4) where θ is prewhiening parameer and R (τ) is normalized block Toepliz marix given by Eq. (3) and I is he ideniy marix. Lengh of he filer The sabiliy of he block Toepliz marix associaed wih differen filer lenghs has been sudied by analyzing is condiion number. A well-

3 SRIVASTAVA & DIMRI: PROCESSING OF SEISMIC DATA 217 esablished fac is ha he error energy associaed wih he design of opimum filer decreases wih an increase in he filer lengh. In oher words, we can expec he soluion of he problem o be well posed by increasing he lengh of he filer. However, is accuracy needs o be esed and we shall demonsrae i hrough a numerical example. in boh cases. However, reducion in condiion number is more in he case of geomeric mean as compared o ha of he arihmeic mean. Finally, he effec of he filer lengh i.e. he size of he block Toepliz marix on he condiion number is analyzed. The condiion number for differen filer lenghs has been compued (Table 1), which shows Applicaion o Daa and Resuls Field daa A six-channel unprocessed marine seismic daa colleced from he wesern coninenal margin of India 16 has been aken for he sudy (Fig.1). The marix of auo and cross correlaion associaed wih he design of deconvoluion operaors for he wo inpu races has been obained. The condiion number for he above block Toepliz marix has been compued firs by normalizaion of he marix using Eq. (3). The condiion number has been obained for boh he cases i.e wih normalizaion and wihou normalizaion and i is observed ha he condiion number has reduced by a process of normalizaion (Figs 2 and 3). The effec of prewhiening parameer aken as a few per cen of arihmeic or geomeric mean on he condiion number has also been deal wih. The condiion number has been obained for he normalized block Toepliz marix adding prewhiening parameer (Eq. 4). The prewhiening parameer has been aken as 1 o 20% of he arihmeic or geomeric mean of he main diagonal of he marix and Figs 2 and 3 show he plo of condiion number versus prewhiening parameer for he wo cases. I is furher observed ha wih an increase of prewhiening parameer he condiion number reduces Fig. 2 Condiion number vs prewhiening parameer in he case of arihmeic mean for synheic daa (a) wih normalizaion and (b) wihou normalizaion Fig. 1 Six channels unprocessed marine seismic races Fig. 3 Condiion number vs prewhiening parameer in he case of geomeric mean for synheic daa (a) wih normalizaion and (b) wihou normalizaion.

4 218 INDIAN J. MAR. SCI., VOL. 33, No.3, SEPTEMBER 2004 Table 1 Condiion number for differen filer lenghs in case of marine seismic daa Filer lengh Wih prewhiening Condiion number Wihou prewhiening Fig. 4 Synheic seismic race divided ino wo secions ha he condiion number has increased wih an increase in he filer lengh. Synheic example A ime series can be represened by he convoluion of waveles and noise series i.e. he Woldian decomposiion heorem. Using his concep 17 synheic seismic race is generaed which is mahemaically represened by X ( ) = α A( ) + k β B( ) (5) where α is he source wavele, β is he noise wavele and 1/k is he signal o noise raio of he race respecively. A() is he refleciviy series generaed using random numbers beween [0,1] and B() is he random Gaussian noise series. The choice of he waveles depends on he naure of he race ha one would require. In several seismic exploraions he Ricker wavele is generally used 18. Synheic races have also been generaed using he minimum delay wavele for he source funcion. Hence, in his sudy he synheic seismic races have been obained for a given se of conrolling parameers, i.e., α = 5.,4.,3.,-1.,0.5. β = 4.,-2.,1.,-1.,0.5 k = 1/1.5 Figure 4 shows he plo of he race ha has been obained and has been divided ino wo secions as repored earlier 5. For hese wo secions he compuaion of he filer coefficiens requires he block Toepliz marix 4. The procedure 4 as discussed has been followed and for differen lenghs of he filer he error energy or he filer performance has been compued 3. Afer normalizing he block Toepliz marix and adding some prewhiening he condiion number has been obained. The resuls are ploed in Fig. 5. From he figure i is clear ha as he lengh of he filer increases, he error energy decreases bu he Fig. 5 Plo of (a) filer lengh versus normalized error energy (b) filer lengh versus condiion number for he synheic example. condiion number is increasing. We see ha for a filer lengh of 11 he wo curves are cuing each oher. The lengh of he filer for which he error energy becomes zero has been given earlier 4. For his example, if we were o look for accurae filers we would need o compue a filer of lengh 19. However, since we need o make a compromise he inersecion poin could be assumed o be an opimum lengh ha akes care of he sabiliy and accuracy of he filer design. Discussion In his sudy we have considered a six-channel marine seismic daa and synheic seismic races and for achieving he sabiliy of he deconvoluion operaors. Single channel secioned inpu and mulichannel seismic deconvoluion involve he inversion of he block Toepliz marix. To obain a sable soluion, he block Toepliz marix associaed wih he deconvoluion operaors has been normalized

5 SRIVASTAVA & DIMRI: PROCESSING OF SEISMIC DATA 219 o ge low values of he condiion number as his measures he well posedness of he problem. Furher, he sabiliy of he algorihm has been achieved by adding he prewhiening parameer. I is well known ha he prewhiening parameer prevens he inverse from blowing. However, i is observed ha wih an increase in he prewhiening parameer he condiion number decreases and also i has been found ha he prewhiening parameer compued from he geomeric mean as compared o he arihmeic mean of he elemens of he main diagonal of he block Toepliz marix reduces he condiion number furher. We have also seen ha wih an increase in he filer lengh he condiion number also increases. However, i is well known ha wih an increase in filer lengh he error energy decreases hereby improving he performance of he filer 2. A rade off beween well posedness and accuracy of he problem exiss and one needs o be careful in choosing he lengh of he filer in such a way ha sabiliy as well as accuracy is mainained. This seismic daa has been used o remove he muliples using he adapive deconvoluion 19. The combined convergence facor for he adapive algorihm is obained. The auo and cross correlaion marix has been sabilized using he same procedure o obain he maximum eigen value, ha is used o obain he convergen facor of he adapive algorihm. The procedure can be applied in various geophysical sudies bu care should be aken for appropriae compromise beween he sabiliy and accuracy of he esimaed parameer. This sudy will be very useful in designing deconvoluion operaors for a single channel secioned inpu/muli channel daa ha will accoun for is sabiliy, accuracy and well posedness. References 1 Robinson E A & Durrani T S, Geophysical signal processing (Prenice-Hall Inc, Englewood Cliffs, NJ), 1986, pp Dimri V P, Deconvoluion and inverse heory (Elsevier Science Publishers, Amserdam), 1992, pp Dimri V P, On he ime-varying Wiener filer, Geoph Pros, 34 (1986) Dimri V P & Srivasava K, Ideal performance crieria for deconvoluion operaors, Geoph Pros, 35 (1987) Dimri V P & Srivasava K, The opimum gae lengh for he ime-varying deconvoluion operaor, Geoph Pros, 38 (1991) Treiel S, Principles of digial mulichannel filering, Geophysics, 35 (1970) Wang R J & Treiel S, The deerminaion of digial Wiener filers by means of gradien mehods, Geophysics, 38 (1973) Treiel S & Lines L R, Linear inverse heory and deconvoluion, Geophysics, 47 (1982) Treiel S & Wang R J, The deerminaion of digial Wiener filers from an ill-condiioned sysem of normal equaions, Geoph Pros, 24 (1976) Hsu K & Baaggeroer A B, Applicaion of he maximumlikelihood mehod (MLM) for sonic velociy logging, Geophysics, 51 (1986) Bickel S H & Marinez D R, Resoluion performance of Wiener filers, Geophysics, 48 (1983) Inman, J R, Resisiviy inversion wih ridge regression, Geophysics, 40 (1975) Lines, L R & Treiel S, A review of leas-squares inversion and is applicaion o geophysical problems, Geoph Pros, 32 (1984) Ursin, B & Zheng Y, Idenificaion of seismic reflecions using singular value decomposiion, Geoph Pros, 33 (1985) Ursin, B, Mehods for esimaing he seismic reflecion response, Geophysics, 62 (1997) Reddy, S I, Roy Chowdhary K, Drollia R K, Ashalaha B, Mial G S, Subrahmanyam C S & Singh R N, On he srucure of he wesern coninenal margin off Mangalore coas, India, J Ass Expl Geoph, 9 (1988) Dash Bibhu P & Obaidullah K A, Deerminaion of signal and noise saisics using correlaion heory, Geophysics, 35 (1970) Ricker N, Wavele conracion, wavele expansion, and he conrol of seismic resoluion, Geophysics, 18 (1953) Srivasava K, An adapive scheme for processing mulirace marine seismic daa, Indian J Mar Sci, 29 (2000)