Visco-acoustic Reverse-time Migration Using the Pseudo-analytical Method

Transcription

1 Visco-acoustic Reverse-time Migratio Usig the Pseudo-aalytical Method J. Ramos-Martiez* (Petroleum Geo-Services), A.A. Valeciao (Petroleum Geo-Services) & M. Orlovich (Petroleum Geo-Services) SUMMARY We derive a ew reverse-time migratio solutio that uses a pseudo-aalytical, two-way extrapolator i order to compesate for atteuatio losses i visco-acoustic media. The algorithm permits large time steppig ad shares the advatages of spectral methods i allowig coarse computatioal grids. The extrapolatio accouts for spatial variatios i atteuatio by usig a series expasio approximatio of the equatios for costat atteuatio. We apply the ew scheme to a dual-sesor field survey from the North Sea, i which the resolutio of the images is clearly improved after compesatig for atteuatio.

2 Itroductio Seismic waves atteuate as they travel through the Earth. This results i a biased loss of highfrequecy eergy, as well as a frequecy-depedet variatio i phase (dispersio). Therefore, to accurately perform imagig i realistic media, the amplitude loss ad the phase distortio should be compesated i the migratio operators. This class of operators solves the visco-acoustic wave equatio, ad requires a extra parameter (the quality factor Q) i order to accurately model the waves propagatig i a atteuatig media. Two mai wave equatio migratio methods ca be used to compesate for visco-acoustic effects i depth migratio. The first method uses oe-way, wave-equatio operators to perform depth extrapolatio i the frequecy domai. The frequecy domai eables atteuatio by turig the phase velocity ito a complex quatity (Valeciao et al., 0). The secod method solves the visco-acoustic, two-way wave equatio i the time domai (e.g., Deg ad McMecha, 008). This solutio is based o a superpositio of mechaical elemets that are approximatios to the costat-q behavior defied withi the frequecy bad of the data (e.g., Emmerich ad Kor, 987). The more elemets used i the superpositio, the more accurate the approximatio, at the expeses of more computatio. O the other had, a more accurate time- domai approach based o the exact, costat-q behavior (e.g., Kjartasso, 979) requires the use of fractioal time derivatives. The accurate computatio of the fractioal derivatives requires the use of large memory resources. To overcome this problem, Che ad Holm (004) itroduced the first algorithm to simulate atteuatio from fractioal wave equatios by usig a fractioal Laplacia that ca be computed i the Fourier domai. The iitial approximatios usig this cocept (e.g., Carcioe, 00) did ot split the effects of dispersio ad absorptio ito differet operators - a algorithmic feature that ca be coveiet for a practical migratio implemetatio. Recetly, Zhu ad Harris (04) derived a wave equatio which splits those terms, ad which Zhu et al. (04) solved by the pseudo-spectral approach withi a reverse time migratio algorithm. Here, we itroduce atteuatio compesatio i RTM by usig a pseudo-aalytical extrapolatio (Etge ad Bradsberg-Dahl, 009). Our method is based o the two-way wave equatio derived by Zhu et al. (04), with the itroductio of a approximatio to the dispersive term of the equatio that accurately simulates the visco-acoustic propagatio i heterogeeous media. As a cosequece, we obtaied a efficiet algorithm that allows the use of coarse spatial grids ad large time steps without jeopardizig the accuracy of the wavefield propagatio. We show a example of the successful applicatio of this ew scheme usig field data from the North Sea i which the image resolutio is clearly improved after compesatig for atteuatio. Two-wave wave equatio i a visco-acoustic media ad Q-RTM implemetatio Assumig the costat-q dispersio relatio itroduced by Kjartasso (979), Zhou ad Harris (04) derived the two-way wave equatio for visco-acoustic, isotropic, heterogeeous media with costat desity as: c γ γ / ( ) σ ( x, τ ( ) σ ( x, σ ( x, = η( 0 t t, () γ γ where the coefficiets as a fuctio of space are defied as η( = c 0 ω0 cos( πγ ), γ γ τ = c 0 ω0 si( πγ ), where γ = π ta ( / Q ). Q is the quality factor, ad c 0 ad ω 0 are the referece velocity ad frequecy, respectively. The advatage of eq. () for migratio is that i the right-had term, the dispersio (firs ad atteuatio (secod) operators are split. However, a drawback is foud i the dispersive operator computed i the waveumber domai for a atteuatig media with spatially variable Q. I this case, there is a ambiguity regardig the value of γ i the

3 expoet that should be used to compute the operator i the waveumber domai. Zhu ad Harris (04) proposed that γ should be computed from a average of the differet Q values of the model. However, the selectio of a uique γ compromises the accuracy of the arrival times of the differet evets i a Q-variable media. We overcome this problem with a series expasio of the operator for dispersio that allows a better approximatio for propagatio i heterogeeous media. We begi with the frequecy (ω) waveumber (k) versio of eq. () ω c η γ ( k ) ( iω) τ ( k ) γ / Assumig that a fuctio of ay variable (γ i this case), ca be approximated aroud a certai value of γ=γ 0 as L( γ ) L( γ 0) ΔL( γ ), (3) it is easy to prove that the first term i the right-side of eq. () ca be approximated aroud γ=0 as η γ ( k ) η( k ) ηγ ( k )[ log( k )] Thus, the variability of the pseudo-laplacia operator is applied i the space domai. (). (4) k as a fuctio of γ is elimiated. Istead, it Crawley et al. (00) showed that the pseudo-aalytic method allows the use of coarse time ad space discretizatio without compromisig the accuracy of the modeled wavefield, eve after log propagatios i time ad space. Sice the first-order time-derivative i the secod term of eq. () affects oly the amplitude of the wavefield, the pseudo-aalytic form of the Laplacia operator is still valid to correct for the error icurred whe the secod-order time derivative is approximated with a secod-order fiite-differece. The ew equatio i its time-discretized, pseudo-aalytic form is FT σ o = σ σ c η( γ FT c ( ) ( ) o x τ x FT { ~ co η( FT f ( k) σ ( k, } {[ ~ ( )log( ~ f k f ( k) )] σ ( k, } ~ ( ~ γ / f ( k) ) σ ( k, FT f ( k) γ / { } {( ) σ ( k, } where stads for the iverse Fourier Trasform ad the ormalized pseudo-aalytic Laplacia (Chiu ad Stoffa, 0) is defied as ~ cos( c0δt k ) f ( k) = (6) c Δt k 0 For the QRTM implemetatio, we follow the work of Zhu et al. (04), which shows a detailed discussio of the mai steps to cosider for the correct compesatio for Q. Oe of the advatages of visco-acoustic wave equatios i the form show i Equatios () ad (5) is that the same algorithm ca be used durig the forward modelig ad RTM extrapolatio, with oly a chage of sig i the amplitude-loss term. Visco-acoustic propagatio Figure shows four paels (z, split by the dark lies correspodig to a wave frot propagated through four differet types of media: (a) acoustic, (b) amplitude-loss oly, (c) dispersive oly, ad (d) visco-acoustic. The referece frequecy is higher tha the frequecy cotet of the source wavelet used i the simulatios. As a cosequece, the propagated wavefields i the dispersive media (Figures. c ad d) travel with a slower velocity tha the backgroud velocity; i.e., they have traveled a shorter distace with respect to the acoustic ad amplitude-loss oly media, for the same travel time. (5)

4 Q-RTM field data example We applied the Q-RTM to a 3D field dataset that was acquired with dual-sesor techology, i the North Sea. The surveyed area is characterized by gas clouds that limit the resolutio of the images at the depth of the reservoir. Valeciao ad Chemigui (03) derived a Q tomographic model, ad used it for atteuatio compesatio i a oe-way wave-equatio migratio (WEM) (Valeciao et al., 0). Figure shows the performace of our RTM algorithm, with ad without Q compesatio, for a maximum frequecy of 50 Hz. As show, Q-compesatio greatly improves the cotiuity ad resolutio at the reservoir level. This is cofirmed by observig the amplitude spectra of the images after depth to time coversio i Figure 3. It shows that the algorithm recovers the high-frequecy cotet of the image, whe compared to that image without Q-compesatio. Figure. Sapshots correspodig to the same travel time, for wavefrots travelig through differet media: (a) Acoustic, (b) amplitude-loss oly, (c) dispersive oly ad (d) visco-acoustic. Red dash lie correspods to the referece wavefrot i the acoustic case. Coclusios We preset a reverse-time, migratio algorithm that uses a pseudo-aalytical extrapolator i order to solve a visco-acoustic, two-way, wave equatio. Our extrapolator icludes a ew approximatio of the dispersive term of the visco-acoustic operator. It allows for a better approximatio of the waves propagatio i heterogeeous atteuatig media. The umerical solutio eables efficiet computatio without compromisig the accuracy of the simulated wavefields, eve for log propagatios i time ad space. Sigificat improvemets i image quality ca be achieved by icorporatig atteuatio i the migratio. This is demostrated by the migratio of a dual-sesor, towed-streamer survey from the North Sea. Ackowledgemets We appreciate the ivaluable discussios with Sea Crawley for the Q-RTM implemetatio, ad the assistace of Boris Tsimelzo ad Elea Klochikhia i the preparatio of this paper. We also thak Steve Kelly, Da Whitmore, Nizar Chemigui ad Sverre Bradsberg-Dahl for helpful suggestios. Refereces Carcioe, J.M. [00]. A geeralizatio of the Fourier pseudospectral method, Geophysics, 75, 6, A- 53-A56. Che, W. ad Holm, S. [004]. Fractioal Laplacia time-space models for liear ad o-liear lossy media exhibitig arbitrary frequecy power-law depedecy, J. Acoust. Soc Am. 5, Chiu, C. ad Stoffa, P.L [0]. Applicatio of ormalized pseudo-laplacia to elastic wave modelig o staggered grids. Geophysics, 76,, T3-T.

5 Crawley, S., Bradsberg-Dahl, S., McMclea, J. ad Chemigui, N. [00] TTI reverse-time migratio usig the pseudo-aalytic method. The Leadig Edge, 9, Deg, F. ad McMecha, G.A. [008] Viscoelastic true-amplitude prestack reverse-time depth migratio, Geophysics, 73,4, S43-S55. Emmerich, H. ad Kor, M. [987]. Icorporatio of atteuatio ito time-domai computatios of seismic wavefields, Geophysics, 5,5-64. Etge, J. ad Bradsberg-Dahl, S. [009] The pseudo-aalytical method: Applicatio of pseudo- Laplacias to acoustic ad acoustic aisotropic wave propagatio: 79th Aual Iteratioal Meetig, SEG, Expaded Abstracts, Kjartasso, E. [979]. Costat-Q wave propagatio ad atteuatio, Joural of Geophysical Research, 84, Valeciao, A. A. Chemigui, N., Whitmore, D ad Bradsberg-Dahl, S.[0] Wave equatio migratio with atteuatio ad aisotropic compesatio, 8st Aual Iteratioal Meetig, SEG, Expaded Abstracts, 3-36 Valeciao, A.A. ad Chemigui, N. [03] Tomography ad migratio i viscoacoustic media,, Thirteeth Iteratioal Cogress of Brazilia Geophysical Society, Zhu, T. ad Harris, J.[04] Modelig acoustic wave propagatio i heterogeeous atteuatig media usig decoupled fractioal Laplacias, Geophysics, 79, T05-T6. Zhu, T., Harris, J. ad Biodi, B. [04]. Q-compesated reverse-time migratio, Geophysics, 79, S77-S87. Figure. RTM Migrated images: without Q compesatio (lef ad with Q compesatio (righ. Figure 3. Amplitude spectra for the RTM images (show i Figure ) after depth to time coversio: without (red) ad with (blue) Q-compesatio.