Recursive integral time extrapolation methods for scalar waves Paul J. Fowler*, Xiang Du, and Robin P. Fletcher, WesternGeco

Transcription

1 Reursive integral time extrapolation methods for salar waves Paul J. Fowler*, Xiang Du, and Robin P. Flether, WesternGeo Summary We derive and ompare a variety of algorithms for reursive time extrapolation of salar waves using approximate operators derived from integral solutions of a wave equation. These methods fall into two ategories: those based on ombining or interpolating homogeneous solutions, and those based on series expansions of heterogeneous operators. The former suffer from osillatory noise at large veloity disontinuities unless the time step is small, whereas the latter allow aurate extrapolation at large time steps, but are more ostly. Introdution Reverse time migration requires high-auray time extrapolation of wavefields. Conventional expliit finitedifferene or pseudospetral methods for time extrapolation of waves are usually limited to small time steps by numerial dispersion errors and instability. Reently, several alternative approahes have been proposed for solving salar wave equations based on integral mixeddomain spae / wavenumber time extrapolation methods (Zhang et al., 007; Soubaras and Zhang, 008; Wards et al., 008; Zhang and Zhang, 009; Etgen and Brandsberg- Dahl, 009; Pestana and Stoffa, 009; Stoffa and Pestana, 009; Liu, et al., 009; Du, et al., 010). We refer here to this entire family of approahes generially as reursive integral time extrapolation (RITE), as they are all based on formal integral solutions of a wave equation. These integral solutions are then applied reursively to extrapolate the wavefield forward or bakwards in disrete time steps. These RITE approahes attempt to ahieve stable, dispersion-free extrapolation in heterogeneous media for large time steps, up to the yquist limit or beyond. We ategorize and ompare here a variety of RITE methods within a ommon derivation framework. RITE methods are oneptually based on a heuristi of time extrapolating a wavefield using multipliation by a phase i t shift of the form e ω, where t is time and ω has units of frequeny and is derived from an appropriate dispersion relation. An analogous onept of spatial wavefield extrapolation using multipliation by a phase shift of the ikz z form e, where z is depth and k z is vertial wavenumber, has been widely used previously as a basis for downwardontinuation modeling and migration methods (e.g., Gazdag, 1978). Many RITE algorithms resemble formal translations of existing downward-ontinuation methods. RITE operators for salar aousti waves The onstant-density aousti salar wave equation an be written as: + A u( t) = 0. (1) t The operator A is defined in the spae domain as A[ u( x, t)] = v (x) u( x, t), () or in the wavenumber domain as A[ u( k, t)] = dk v (k k) k u( k, t). (3) A formal integral time extrapolation operator for the salar wave equation 1 an be written as iφ t u( t + t) = e u( t), (4) where Φ A. This form of the extrapolator requires omplex-valued data. Adding solutions from equation 4 for u( t + t) and u( t t) gives a time extrapolator for real-valued data of the form u( t + t) = os( Φ t) u( t) u( t t). (5) We refer to extrapolators of the form of equation 4 as onestep methods, and those from equation 5 as two-step methods. Four-step or higher in time methods an be developed similarly. The square-root, exponential, and osine funtions of operators indiated in equations 4 and 5 an be given formal meaning by deomposition into eigenvetors and eigenvalues of the wavenumber-domain operator A defined in equation 3. This in priniple gives integral solutions for free-spae time extrapolation of the salar wave equation 1 that will be aurate in heterogeneous media for arbitrarily large time steps. This is a time extrapolation analog of the eigen-deomposition downward-ontinuation approah disussed by Jaobs (1980), Pai (1985), and Kosloff and Kessler (1987). However, eigen-deomposition of the operator A is omputationally too expensive for realisti three-dimensional models. The various approahes suggested for pratial approximation of the heterogeneous RITE operator in equation 4 an be divided into two basi families: those based on ombination or interpolation of loally homogeneous solutions, and those based on series expansion approximations of the heterogeneous operator. Methods for ombining homogeneous RITE operators For homogeneous media, the operator A redues to ust A = v k and the one-step extrapolator beomes u k t + t = e u k t iv k t (, ) (, ). (6) 310

2 Reursive integral time extrapolation of waves This operator is now easily implemented in the wavenumber domain, and is essentially free from numerial dispersion, even for arbitrarily large time steps. One simple way to implement a variable-veloity RITE sheme for heterogeneous media would be to design separate onstant-veloity extrapolators for every point in spae, using the partiular veloity appropriate to eah individual loation. We term suh brute fore solutions phase-shift plus seletion (PSPS). Simple PSPS is too expensive to be of muh pratial use. Wards et al. (008) and Etgen and Brandsberg-Dahl (009) suggested less-expensive approahes based on extrapolating a wavefield using only a few disrete onstant veloities, followed by interpolation between these data volumes. We term suh methods phase-shift plus interpolation (PSPI) in analogy with the downward-ontinuation method introdued by Gazdag and Sguazzero (1984). PSPI methods an also be based on the two-step onstantveloity extrapolator u( t + t) = os( v k t) u( t) u( t t). (7) Etgen and Brandsberg-Dahl (009) suggest modifying the osine extrapolator to the form [os( v k t) 1] / ( v t ), alling this normalized operator a pseudo-laplaian. Comparing the Taylor expansions of the operators shows that the pseudo-laplaian normalization anels the leading order veloity dependene. This normalization an thus inrease interpolation auray for small time steps. Zhang and Zhang (009) introdue another sheme for ombining onstant-veloity RITE operators based on earlier downward-ontinuation methods used by Chen and Liu (006). One an build a matrix of one-step onstant-veloity extrapolators mn iv k t m n = e, defined for ordered sets of disrete veloity values { v } and disrete m wavenumber magnitudes { k n }. This matrix an then be deomposed by singular value deomposition into a T produt of three matries of the form =ΞΛΨ. Λ is a diagonal matrix of the orresponding singular values λ. One an then write the elements of matrix as the separable deomposition iv t λ = m n e k λ ξ ( v ) ψ ( k ), (8) = 1 m n where ξ ( v ) = Ξ is a olumn of the matrix Ξ, m m ψ ( k ) = Ψ a olumn of the matrix Ψ, and λ is the n n number of singular values used. In pratie, Zhang and Zhang (009) suggest using the optimal separable approximation (OSA) to reate a parsimonious approximation using only the largest singular values. Given then a method for interpolating the funtions ξ and ψ, this separable deomposition an be extended to represent extrapolators for any value within the desired ranges, yielding an extrapolation sheme of the form { ψ } ( x, ) 1 λ ξ [ ( x)] [ ( k)] ( k, ), (9) u t + t = v FT k u t = 1 where FT indiates a 3-D spatial Fourier transform. Series expansions of heterogeneous RITE operators The approximation methods disussed in the previous setions onstrut a heterogeneous solution using loally homogeneous salar-phase solutions, with no diret use of heterogeneous extrapolators. Series expansions provide one general method for deriving approximations to the wavenumber-onvolutional time extrapolation operators. Series expansions are easier to onstrut for two-step operators, beause only even powers of the phaseφ are needed. Suppose that one has a set of basis polynomials { P ( x) p x } i =. Approximating the two-step osine i = 1 operator as an expansion in these basis polynomials gives where os( Φ t) b P ( Φ t) = a Φ t, (10) a = 1 = 0 = 0 = b p. The series in equation 10 an be i i i evaluated in nested form as a a 1 os( Φ t) a + t + t ( + ). (11) 0 I Φ I Φ I a a 0 1 In this form, evaluation of eah additional term requires multipliation of the data by k in the wavenumber domain, followed by transformation bak to the spae domain and multipliation by the veloity squared, followed by transformation bak to the wavenumber domain, and so forth. ote that trunating suh an evaluation after theφ term yields a onventional seondorder in time, pseudo-spetral in spae algorithm. The exat riterion for operator stability requires omputation of a full eigen-deomposition, whih is usually infeasible. However, a good estimate an be ahieved using the orresponding stability riteria for onstant-veloity operators for the largest values of veloity, wavenumber, and time-step size. For the two-step osine extrapolator, Soubaras and Zhang (008) give the approximate stability riterion that the osine approximation used must satisfy os( φ) 1 on the 311

3 Reursive integral time extrapolation of waves interval [ φ, φ ], where φ = π v t / r and max max max ma x 1/ r = ( x + y + z ). Soubaras and Zhang (008) also present a similar auray riterion, requiring good fitting of the osine up to a value given by φ = π t f. For time stepping at yquist, this redues to ust φ = π. Perhaps the simplest expansion is a Taylor series, but they an onverge very slowly, and so are not omputationally very effiient. Expansions using Legendre or Chebyshev polynomials an provide more effiient and aurate approximations. Very large time steps an be used if enough terms in these expansions are retained. Chebyshev polynomial expansions have been used for downward ontinuation by Kosloff and Kessler (1987) and for time extrapolation by Tal-Ezer et al. (1987), Kosloff et al. (1989), Pestana and Stoffa (009), and Stoffa and Pestana (009). These orthogonal polynomial expansions fit the osine by distributing errors evenly over the entire stability interval. The required auray interval is usually muh smaller. One an instead numerially ompute polynomial oeffiients that optimize the fit over the interval [ φ, φ ], while onstraining the optimization to fore stability over the larger interval. Soubaras and Zhang (008) desribe one suh optimal polynomial two-step RITE approah, using the Remez exhange algorithm to derive optimized oeffiients. This is an extension of the downwardontinuation extrapolators presented by Soubaras (1996). We use here an alternative onstrained weighted leastsquares approah (WLSQ) based on the downwardontinuation method of Thorbeke et al. (004). Examples In Figure 1 we show a series of numerial experiments omparing four different RITE implementations. The model for eah has an upper layer with v=1500 m/s and a lower layer with v=4500 m/s, with an abrupt ump between layers. The spatial sampling is 15 m. A point soure with a maximum frequeny of 50 Hz is loated in the upper layer, and eah image shows a wavefield snapshot at t=0.8s. The olumns show omputations using time-step values t of s, s, 0.01 s, and 0.0 s, equivalent to 0.5, 0.5, 1.0, and.0 times the yquist time step. The first row (a-d) uses a one-step PSPI operator, the seond (e-h) a two-step PSPI operator, the third (i-l) a onestep OSA operator with 6 eigenvalues, and the fourth (m-p) a two-step WLSQ series expansion operator, using 13 terms for the first three images and 18 terms for the last. All the methods give good results for the smallest time step ( t = s ). The one-step PSPI operator in the first row gives good results for time steps up to t = s max but displays osillatory artifats for larger time steps. The two-step PSPI method in the seond row has larger artifats, even for time step t = 0.005s. The OSA results in the third row are nearly idential to the one-step PSPI in the first row. ote here that the OSA and PSPI methods essentially redue to PSPS beause only two veloity values are present, so no atual interpolation is being done. The osillatory artifats seen in the OSA and PSPI results arise from the attempt to ombine homogeneous operators. The two-step WLSQ method in the fourth row avoids these artifats, even for large time steps, by more aurately approximating the orret heterogeneous operator. Further extensions of RITE methods RITE methods an also be implemented in the spae domain only. For PSPS, PSPI, and OSA RITE methods, one an design the extrapolation operators in the wavenumber domain, but transform them bak to beome onvolutional spae-domain operators. For PSPS, this yields an analog of the downward-ontinuation approah of Blaquiere et al. (1988). These spae-domain operators an be preomputed and applied repeatedly at every time step, but the storage requirements an beome very large. Similarly, two-step series expansion methods an be implemented in the spae domain by replaing the wavenumber-domain operator in equation 11 with a finitedifferene spatial Laplaian. Our study here is not exhaustive. Using rational fration expansions instead of series to approximate operators leads to impliit extrapolation shemes (e.g., Claerbout, 1985). For time extrapolation, the resulting sets of impliit equations appear to beome extremely large. Fourier finite-differene (FFD) (Ristow and Ruhl, 1994) and generalized sreen propagator (e.g., Le Rousseau and de Hoop, 1998) methods are based on omputing perturbations from onstant-veloity referene operators (Sava, 000). For time extrapolation, these methods will need to handle a muh larger range of veloities than for downward ontinuation (a problem shared also by PSPI methods), and so an be expeted to be limited to small time steps. However, Song and Fomel (pers. omm.) reently report promising results for a FFD time-stepping method. One an also write a salar equation for P and SV waves in VTI media (Fowler et al., 010), and find formal singlemode exponential operator solutions analogous to equation 4, allowing extension of the various RITE methods disussed here to either P or SV waves in TI media. Etgen and Brandsberg-Dahl (009) disuss TI PSPI methods. Extensions of OSA methods to VTI media were presented by Liu et al. (009) and by Du et al. (010). Two-step osine series expansion extrapolators an also be generalized to VTI or TTI media, but the inrease in omputational ost an beome onsiderable. 31

4 Reursive integral time extrapolation of waves Conlusions The RITE methods we have disussed here allow extension for use in reverse time migration of many of the methods used previously for depth extrapolation. We examined two maor families of mixed-domain RITE methods: ones that ombine homogeneous extrapolators, and ones based on series approximations to heterogeneous extrapolators. The latter allow aurate extrapolation with larger time steps than the former, and exhibit fewer artifats at large abrupt veloity disontinuities. However, this improved auray usually requires muh greater omputational effort per time step, espeially for anisotropi media. a) b) ) d) e) f) g) h) i) ) k) l) m) n) o) p) Figure 1: Wavefield snapshots at time 0.8 s for a two-layer model. The upper layer veloity is 1500 m/s and bottom layer veloity is 4500 m/s. Images from left to right represent time steps of t = s, t = s, t = 0.01 s, and t = 0.0 s. The top row (a-d) uses a one-step PSPI approah. The seond row (e-h) uses a two-step PSPI approah. The third row (i-l) uses a one-step OSA approah. The bottom row (m-p) uses a twostep WLSQ series expansion approah. 313

5 EDITED REFERECES ote: This referene list is a opy-edited version of the referene list submitted by the author. Referene lists for the 010 SEG Tehnial Program Expanded Abstrats have been opy edited so that referenes provided with the online metadata for eah paper will ahieve a high degree of linking to ited soures that appear on the Web. REFERECES Blaquiere, G., H. W. J. Debeye, C. P. A. Wapenaar, and A. J. Berkhout, 1988, 58th Annual International Meeting, SEG, Expanded Abstrats, Chen, J., and H. Liu, 006, Two kinds of separable approximations for the one-way wave operator: Geophysis, 71, no. 1, T1 T5, doi: / Claerbout, J. F., 1985, Imaging the Earth s Interior: Blakwell Sientifi Publiations, In. Du, X., R. P. Flether, and P. J. Fowler, 010, Pure P-wave propagators versus pseudo-aousti propagators for RTM in VTI media: 7nd EAGE Conferene & Exhibition, Extended Abstrats. Etgen, J., and S. Brandsberg-Dahl, 009, The pseudo-analytial method: appliation of pseudo-laplaians to aousti and aousti anisotropi wave propagation: 79th Annual International Meeting, SEG, Expanded Abstrats, Fowler, P. J., X. Du, and R. P. Flether, 010, Coupled equations for reverse time migration in transversely isotropi media : Geophysis, 75, no. 1, S11 S, doi: / Gazdag, J., 1978, Wave equation migration with the phase-shift method: Geophysis, 43, , doi: / Gazdag, J., and P. Sguazzero, 1984, Migration of seismi data by phase shift plus interpolation: Geophysis, 49, , doi: / Jaobs, A., 1980, Wide-angle variable veloity one-way wave-equation modeling: Presented at the 50th Annual International Meeting, SEG. Kosloff, D., and D. Kessler, 1987, Aurate depth migration by a generalized phase-shift method: Geophysis, 5, , doi: / Kosloff, D., A. Q. Filho, E. Tessmer, and A. Behle, 1989, umerial solution of the aousti and elasti wave equation by a new rapid expansion method: Geophysial Prospeting, 37, no. 4, , doi: / tb01.x. Le Rousseau, J. H., and M. V. de Hoop, 1998, Modeling and imaging with the generalized sreen algorithm: 68th Annual International Meeting, SEG, Expanded Abstrats, Liu, F., S. A. Morton, S. Jiang, L. i, and J. P. Leveille, 009, Deoupled wave equations for P and SV waves in an aousti VTI media: 79th Annual International Meeting, SEG, Expanded Abstrats, Pai, D. M., 1985, A new solution method for the wave equation in inhomogeneous media : Geophysis, 50, , doi: / Pestana, R., and P. L. Stoffa, 009, Rapid expansion method (REM) for time-stepping in reverse time migration (RTM): 79th Annual International Meeting, SEG, Expanded Abstrats, Ristow, D., and T. Ruhl, 1994, Fourier finite-differene migration: Geophysis, 59, , doi: /

6 Sava, P. C., 000, A tutorial on mixed-domain wave-equation migration and migration veloity analysis: Stanford Exploration Proet Report, 105, Soubaras, R., 1996, Expliit 3-D migration using equiripple polynomial expansion and Laplaian synthesis: Geophysis, 61, , doi: / Soubaras, R., and Y. Zhang, 008, Two-step expliit marhing method for reverse-time migration: 70th EAGE Conferene & Exhibition, Extended Abstrats. Stoffa, P. L., and R. Pestana, 009, umerial solution of the aousti wave equation by the rapid expansion method (REM) A one-step time evolution algorithm: 79th Annual International Meeting, SEG, Expanded Abstrats, Tal-Ezer, H., D. Kosloff, and Z. Koren, 1987, An aurate sheme for seismi forward modeling: Geophysial Prospeting, 35, no. 5, , doi: / tb00830.x. Thorbeke, J. W., C. P. A. Wapenaar, and G. Swinnen, 004, Design of one-way wavefield extrapolation operators, using smooth funtions in WLSQ optimization: Geophysis, 69, , doi: / Wards, B., G. Margrave, and M. Lamoureux, 008, Phase-shift time-stepping for reverse-time migration: 78th Annual International Meeting, SEG, Expanded Abstrats, Zhang, Y., G. Zhang, D. Yingst, and J. Sun, 007, Expliit marhing method for reverse-time migration: 77th Annual International Meeting, SEG, Expanded Abstrats, Zhang, Y., and G. Zhang, 009, One-step extrapolation method for reverse-time migration: Geophysis, 74, no. 4, A9 A33, doi: /