Th Q Estimation Through Waveform Inversion

Transcription

1 Th Q Esimaion Through Waveform Inversion J. Bai* (ION) & D. Yings (ION) SUMMARY This paper presens an approach o esimae he qualiy facor Q hrough waveform inversion. In a viscoacousic medium consising of one sandard linear solid, sress and srain relaxaion imes govern he dissipaion mechanism. Their difference, normalized o be a uniless variable τ, deermines he magniude of Q. In his paper we ieraively opimize a τ model by minimizing an objecive funcion ha measures he residuals beween recorded and synheic seismic daa. The τ model is hen convered o is corresponding Q model. A viscoacousic Marmousi model demonsraes he accuracy of he approach. For a field daa from he Gulf of Mexico we presen a workflow o esimae is Q model and hen opimize is velociy model hrough waveform inversions wih he aenuaion compensaion. The workflow shows some promise o ge he final seismic producs wih he aenuaion compensaion for physical maerials.

2 Inroducion Because of he conversion of elasic energy ino hea seismic waves are aenuaed and dispersed as hey propagae. This anelasic behavior can resul in ampliude decrease, wavele disorion and he loss of high-frequency componens. I herefore can cause srong fooprins on seismic inversion and imaging, AVO/AVA analysis, and make inerpreaion more difficul. I is herefore imporan o compensae for he aenuaion effecs in he final seismic produc. An aenuaion model, described by he qualiy facor Q, is widely used in various compensaion mehods. Because he aenuaion effecs cause he loss of high-frequency componens, seismic ampliude specra naurally serve as a carrier for Q esimaion. Usually a Q model can be esimaed eiher hrough he ampliude-specra-raio mehod (Dasgupa and Clark, 1998) or hrough he measuremen of he relaive shif of dominan frequency (Quan and Harris, 1997). These mehods assume ha scaering, geomerical spreading, and oher non Q-relaed facors have been removed from seismic daa. Given a Q model, a viscoelasic mechanical model consising of sandard linear solids (SLSs) provides a powerful ool o model real earh maerials (Robersson e al., 1994). One SLS consiss of a spring in parallel wih a spring and a dashpo in series. I can approximae a consan Q wihin a defined frequency band. A series of SLSs conneced in parallel can yield a quie general mechanical viscoelasiciy (Day and Minser, 1984). For each SLS is relaxaion mechanism describes he physical dissipaion on seismic waves. Is sress relaxaion ime τσ and is srain relaxaion ime τε govern his relaxaion mechanism. Their difference, expressed as a uniless variable τ (= τε/τσ -1), deermines Q. Waveform inversion esimaes subsurface parameers in a way ha minimizes he residuals beween he recorded and synheic seismic daa. I is aracive in is abiliy o produce parameer models wih high resoluion for complex geological srucures. Since he gradien-based opimizaion mehods were inroduced (Taranola, 1984), waveform inversion has achieved subsanial success a he esimaion of velociy models (Vigh e al., 2011; Wang e al., 2012) and anisoropic parameers (Prieux e al., 2012) in pracice. This paper inroduces an approach for Q esimaion by waveform inversion. For a viscoacousic model consising of one SLS his approach esimaes a τ model. The τ model is hen convered o is corresponding Q model. We se up an objecive funcion and hen derivae a gradien from i. Viscoacousic wave equaions for forward modeling and is adjoin are used for he calculaion of he objecive funcion and he gradien (Bai e al., 2012). A nonlinear conjugae gradien mehod is employed o updae he τ model. Tes wih a viscoacousic Marmousi model demonsraes ha he approach can produce a complex Q model wih high resoluion. We also presen a workflow comprising a Q inversion, followed by a velociy updae hrough viscoacousic waveform inversion for a 3D deep-waer OBC (ocean boom cable) field daa from he Gulf of Mexico (GOM). This es shows some promise in boh Q esimaion and velociy esimaion wih he aenuaion compensaion for physical maerials. Theory In a viscoacousic model consising of one SLS, viscoacousic wave equaion can be used o simulae he aenuaion effecs on seismic waves during heir propagaion (Bai e al., 2012). 1 2 P v 2 = (1+ τ )ρ ( 1 P) r + f (1) 2 ρ wih τ τ 1 r = [ e σ H( )]*[ ρ ( P)], (2) τ ρ σ

3 where P = P(x,;x s ) is he wavefield a ime and a a posiion x for a source locaed a x s, ρ = ρ(x) is densiy, v = v(x) is velociy, f is source, H() is he Heaviside funcion, and r is a memory variable. The memory variable is a causal ime convoluion and describes he dissipaion mechanism. Is kernel is of exponenial characer. Since i decays, energy is dissipaed. For a fixed frequency, τσ is nearly consan (Robersson e al., 1994). Consequenly he decay speed mainly depends on τ, which deermines he magniude of he qualiy facor Q. Q = ( + 1) 1, (3) τ This relaionship clearly shows ha we can inver a τ model and hen conver i o a Q model. We herefore ieraively opimize a τ model by minimizing a oal value (TV) regularized objecive funcion, which measures he residuals beween he recorded and synheic seismic daa 2 J() τ = Γ( d αd) + λt, (4) 0 wih he TV regularizaion consrain defined as T() τ = ( τ τ ) + β dx, (5) 0 where d 0 = d 0 (x r,;x s ) is he recorded daa and d = d(x r,;x s ) is he synheic daa a he receiver posiion x r. α (= <d, d 0 >/ d 2 ) is a normalizaion scale. The operaor Γ is a precondiioning operaor on he residual d 0 -αd. λ is a weighing scale. The small value β is squared and added o he square norm of gradien beween he curren τ and he original τ 0 o void singulariy of he gradien of T. The gradien for τ updae is given by g(x) = 2α x s [ρ 1 ρ (P) 1 τ r]r + λ (τ τ 0 ), (6) (τ τ 0 ) 2 + β 2 where R = R(x,;x s ) is he wavefield obained by applying he adjoin of forward modeling on he residual Γ(d 0 -αd) (Bai e al., 2012). Viscoacousic wave equaions for forward modeling and is adjoin are solved by sable high-order finie-difference schemes in cenered grids. A normalized gradien by he ampliude of forward wavefield acceleraes convergence. g( x) g ( x) =, (7) n P ( x, ; x ) + κ where κ is a whiening facor o avoid singulariy. xs s We updae he τ model by using he Polak-Ribière implemenaion of nonlinear conjugae gradien mehod. A line search uses he BB formula (Barzilai and Borwein, 1988) for an iniial esimae of sep lengh. The BB formula is an efficien way o esimae a sep lengh for he TV regularized problem. I does no require exra forward modeling for he evaluaion of objecive funcion. Examples We firs use a viscoacousic Marmousi model o demonsrae our mehod for Q esimaion. The model includes a waer layer from surface down o 500 m. A Q model shown in Figure 1 is direcly mapped from is velociy model. The aenuaion in waer is weak (Q = 5000) while i is srong below waer since Q ranges from 20 o 80. Using he velociy and Q models, we generae a viscoacousic synheic daase for a consan densiy. The daase has 125 shos. Sho inerval is 100 m. Each sho has 161 receivers. Receiver inerval is 20 m. We sar waveform inversion from a consan Q model (Q = 5000) and only use frequencies below 9 Hz. A muli-scale approach is carried ou from low o high frequencies o bypass boh local-minima and cycle-skipping problems. In he inversion he rue velociy model is used. The waveform inversion evenually generaes a high-resoluion Q model shown in Figure 1. The invered model reveals complex Q anomalies in deails.

4 We nex consider a 3D OBC field daa from he deep-waer GOM. The daase has oally shos. Each sho has 239 receivers. We es a workflow o opimize a velociy model hrough waveform inversions wih he aenuaion compensaion. In he inversions he daa wih offse range from 3500 m o 6500 m are used and frequencies range from 2 o 9 Hz. Firs our sraegy relies on he consrucion of a Q model. We inver he Q model (Figure 2) from a consan Q model (Q = 5000). The high-value Q means no aenuaion a he beginning. The invered Q model indicaes srong aenuaion in some areas. A velociy model (Figure 3) is kep consan in he firs waveform inversion. Nex we keep he invered Q model consan and opimize he velociy model by a viscoacousic waveform inversion. We only updae sedimen velociy. The invered velociy model is shown in Figure 3. The seismic aenuaion is srong in he area of ineres (Figure 4) where he geological srucures are poorly imaged (Figure 4). The image is improved by using he invered velociy model (Figure 4(c)). Energy is beer focused in he area of ineres. This example shows ha incorporaing aenuaion in model building is helpful o improve migraion images in pracice. Figure 1 Viscoacousic Marmousi model. The rue Q model. The invered Q model from waveform inversion. Conclusions In his paper we expand he applicaion of waveform inversion on Q esimaion. The approach uses raw seismic daa wihou removing scaering, geomerical spreading, and oher non Q-relaed facors. This makes he approach robus and reliable for real daa. The Marmousi model demonsraes ha a Q model can be accuraely esimaed when is rue velociy model is used. The invered Q model has high resoluion o reveal complex Q anomalies in deails. Using he GOM field daa, we presen a workflow o esimae a Q model and hen opimize a velociy model hrough waveform inversions wih he aenuaion compensaion. The invered velociy model reduces he aenuaion fooprins in he final image. Acknowledgemens We hank ION Geophysical for he permission o publish his work. We also hank our colleagues in ION Geophysical for heir discussions. Thanks go o IFP for he Marmousi model. References Bai, J., D. Yings, R. Bloor, and J. Leveille, 2012, Waveform inversion wih aenuaion: SEG Technical Program Expanded Absrac. Barzilai, J., and J. Borwein, 1988, Two-poin sep size gradien mehods: IMA Journal of Numerical Analysis, 8, Dasgupa, R., and R. A. Clark, 1998, Esimaion of Q from surface seismic reflecion daa: Geophysics, 63, Day, S. M. and Minser, J. B., 1984, Numerical simulaion of wave fields using a Padé approximan mehod: Geophys. J. R. Asr. Soc., 78,

5 Prieux, V., R. Brossier, S. Opero, J. Virieux, O.I. Barkved and J.H. Kommedal, 2012, Twodimensional anisoropic visco-elasic full waveform inversion of wide-aperure 4C OBC daa from he Valhall Field: EAGE expended absrac. Robersson, J. O. A, Blanch J. O., and Symes W. W., 1994, Viscoelasic finie-difference modeling: Geophysics, 59, Taranola, A., 1984, Inversion of seismic reflecion daa in he acousic approximaion: Geophysics, 49, Quan, Y., and J. M. Harris, 1997, Seismic aenuaion omography using he frequency shif mehod: Geophysics, 62, Vigh, D., J. Kapoor, and H. Li, 2011, Full-waveform inversion applicaion in differen geological seings: SEG Technical Program Expanded Absrac. Wang, C., Yings D., Bloor R., and Leveille J., 2012, Applicaion of VTI waveform inversion wih regularizaion and precondiioning o real 3D daa: EAGE Expanded Absrac. Figure 2 The invered Q model for he 3D GOM field daa. sal (c) sal Figure 4 The area of ineres shown in he invered Q model. Migraion images obained from he iniial velociy model and (c) he invered velociy model from viscoacousic waveform inversion. Figure 3 The 3D GOM field daa example. The iniial velociy model. The invered velociy model from waveform inversion.