One-Way and One-Return Approximations (DeWolf Approximation) For Fast Elastic Wave Modeling in Complex Media

Transcription

1 One-Way and One-Return Approximations (DeWol Approximation) For Fast Elastic Wave Modeling in Complex Media Ru-Shan Wu, Xiao-Bi Xie and Xian-Yun Wu Modeling and Imaging laboratory, Institute o Geophysics and Planetary Physics, University o Caliornia, Santa Cruz Summary he De Wol approximation has been introduced to overcome the limitation o the Born and Rytov approximations in long range orward propagation and backscattering calculations. he De wol approximation is a multiple-orescattering-single-backscattering (MFSB) approximation, which can be implemented by using an iterative marching algorithm with a single backscattering calculation or each marching step (a thin-slab). hereore, it is also called a one-return approximation. he marching algorithm not only updates the incident ield step-by-step, in the orward direction, but also the Green s unction when propagating the backscattered waves to the receivers. his distinguishes it rom the irst order approximation o the asymptotic multiple scattering series, such as the generalized Bremmer series, where the Green s unction is approximated by an asymptotic solution. he De Wol approximation neglects the reverberations (internal multiples) inside thin-slabs, but can model all the orward scattering phenomena, such as ocusing/deocusing, diraction, reraction, intererence, as well as the primary relections. In this chapter, renormalized MFSB (multiple-orescattering single-backscattering) 1

2 equations and the dual-domain expressions or scalar, acoustic and elastic waves are derived by using a uniied approach. wo versions o the one-return method (using MFSB approximation) are given: one is the wide-angle (compared to the screen approximation, no small-angle approximation is made in the derivation), dual-domain ormulation (thin-slab approximation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation or the wave-medium interaction, it can be clearly seen that the orward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or relected waves, are mainly controlled by impedance perturbations. Later in this chapter the validity o the thin-slab and screen methods, and the wide-angle capability o the dual-domain implementation are demonstrated by numerical examples. Relection coeicients o a plane interace, derived rom numerical simulations by the wide-angle method, are shown to match the theoretical curves well up to critical angles. he methods are applied to the ast calculation o synthetic seismograms. he results are compared with inite dierence (FD) calculations or the elastic French model. For weak heterogeneities ( ± 15% perturbation), good agreement between the two methods veriies the validity o the one-return approach. However, the one-return approach is about -3 orders o magnitude aster than the elastic FD algorithm. he other example o application is the modeling o amplitude versus angle (AVA) responses or a complex reservoir with heterogeneous overburdens. In addition to its ast computation speed, the one return method (thin-slab and complex-screen propagators) has some special advantages when applied to the thin-bed and random layer responses. Keywords: one-way wave equation, generalized screen propagator, seismic wave

3 modeling, AVO. 1. INRODUCION One-way approximation or wave propagation has been introduced and widely used as propagators in orward and inverse problems o scalar, acoustic and elastic waves (e.g., Claerbout, 197, 1976; Landers and Claerbout, 197; Flatté and appert, 1975; Corones, 1975; appert, 1977; McCoy, 1977; Hudson, 198; Ma, 198; Wales and McCoy, 1983; Fishman and McCoy, 1984, 1985; Wales, 1986; McCoy and Frazer, 1986; Collins, 1989, 1993; Collins and Westwood, 1991; Stoa et al., 199; Fisk and McCarter, 1991; Wu and Huang, 199; Ristow and Ruhl, 1994; Wu, 1994, 1996, 3; Wu and Xie, 1994; Wu and Jin, 1997; Grimbergen et al., 1998; Stralen et al., 1998; Wild and Hudson, 1998; homson, 1999, 5; De Hoop et al., ; Lee at al., ; Wild et al., ; Wu et al., a,b; Le Rousseau and de Hoop, 1; Wu and Wu, 1; Xie and Wu, 1, 5; Han and Wu, 5). he great advantages o one-way propagation methods are the ast speed o computation, oten by several orders o magnitudes aster than the ull wave inite dierence and inite element methods, and the huge saving in internal memory. he successul extension and applications o one-way elastic wave propagation methods has stimulated the research interest in developing similar theory and techniques or relected or backscattered wave calculation. here are several approaches in extending the one-way propagation method to include backscattering and multiple scattering calculations. he key dierence between these approaches is how to deine a reerence Green s unction or constructing one-way propagators. he generalized Bremmer series (GBS) approach (Corones, 1975; De Hoop, 1996; Wapenaar, 1996, 1998; van Stralen et al., 1998; homson, 1999; Le Rousseau and de 3

4 Hoop, 1) adopts an asymptotic solution o the acoustic or elastic wave equation in the heterogeneous medium as the Green s unction, i.e. the one-way propagator in the preerred direction. he multiple scattering series is based on the interaction o Green s ield (incident ield) with the medium heterogeneities. he other approach, i.e. the generalized screen propagator (GSP) approach (Wu, 1994, 1996, 3; Wu and Xie, 1993, 1994; Wu et al., 1995; Wild and Hudson, 1998; de Hoop et al., ; Wild et al., ; Xie et al., ; Xie and Wu, 1, 5), on the other hand, does not use asymptotic solutions. Instead, the approach uses the multiple-orward-scattering (MFS) corrected one-way propagator as the Green s unction. When the backscattered ield, calculated at each thin-slab, is propagated to the backward direction, the same MFS corrected one-way propagator is used. In surace wave modeling, Friederich et al. (1993) and Friederich (1999) adopted a similar approach o MFS approximation (See Chapter by Maupin in this book). However, the approach did not apply the MFS correction to the one-way propagator in the backward direction or the backscattered waves, and thereore did not take the ull advantages o the De Wol approximation. In section, we will compare GBS and GSP approaches ater introducing the De Wol multiple scattering series and the related approximation. In the rest o this chapter we will concentrate on the ormulation and applications o the generalized screen approach. In the generalized screen approach, Wu and Huang (1995) introduced a wide-angle modeling method or backscattered acoustic waves using the multiple-orward-scattering approximation and a phase-screen propagator. Xie and Wu (1995; 1) extended the complex screen method to include the calculation o backscattered elastic waves under the small-angle approximation. Wu (1996) derived a more general theory or acoustic and elastic 4

5 waves using the De Wol approximation, and the theory provided two versions o algorithms: the thin-slab method and the complex-screen method. Later, Wu and Wu (1999, 3a) introduced a ast implementation o the thin-slab method and a second order improvement or the complex-screen method. In section 3, the dual-domain thin-slab ormulations or the case o scalar, acoustic and elastic media are derived. In section 4, a ast implementation o the thin-slab propagator is presented with numerical examples. he excellent agreement between the thin-slab and the elastic FD method in the numerical examples demonstrates the validity and eiciency o the theory and method. In section 5, the small-angle approximation is introduced to derive the screen approximation, which is less accurate or wide angle scattering but is more eicient than the thin-slab method. he validity and potential o the one return approach and the wide-angle capability or the dual-domain implementation are demonstrated by numerical examples or both thin-slab and screen methods applying to the calculation o synthetic seismic sections. In section 6, the thin-slab method is applied to wave ield and amplitude versus oset (AVO) modeling in exploration seismology.. BORN, RYOV, DE WOLF APPROXIMAIONS AND MULIPLE SCAERING SERIES he perturbation approach is one o the well-known approaches or wave propagation, scattering and imaging (see Chapter 9 o Morse and Feshbach, 1953; Chapter13 o Aki and Richards, 198; Wu, 1989). raditionally, the perturbation method was used only or weakly inhomogeneous media and short propagation distance. However, recent progress in this direction has led to the development o iterative perturbation solutions in the orm o a one-way marching algorithm or scattering and imaging problems in strongly heterogeneous 5

6 media. For a historical review, see section 3.1 o Wu (3). In this section, we give a theoretical analysis o the perturbation approach, including the Born, Rytov and De Wol approximations, as well as the multiple scattering series. he relatively strong and weak points o the Born and Rytov approximations are analyzed. Since the Born approximation is a weak scattering approximation, it is not suitable or large volume or long-range numerical simulations. he Rytov approximation is a smooth scattering approximation, which works well or long-range small-angle propagation problems, but is not applicable to large-angle scattering and backscattering. hen, the De Wol approximation (multiple orescattering single backscattering, or one-return approximation ) is introduced to overcome the limitations o the Born and Rytov approximations in long range orward propagation and backscattering calculations, which can serve as the theoretical basis o the new dual-domain propagators..1. Born approximation and Rytov approximation: their strong and weak points For the sake o simplicity, we consider the scalar wave case as an example. he scalar wave equation in inhomogeneous media can be written as ω + u( r) =, (1) c ( r) where ω is the circular requency, r is the position vector, and c( r ) is wave velocity at r. Deine c as the background velocity o the medium, resulting in ( + k ) u ( r ) = k ε ( r ) u ( r ) where k = ω c is the background wavenumber and, () c ε ( r ) = 1 c ( r ) (3) is the perturbation unction (dimensionless orce). Set ( ) u r = u ( r) + U( r), (4) 6

7 where u ( r ) is the unperturbed wave ield or incident wave ield (ield in the homogeneous background medium), and U( r ) is the scattered wave ield. Substitute (4) into () and notice that u ( r ) satisies the homogeneous wave equation, resulting in where g( r;r ') u 3 ( r ) = u ( r ) + k d r ' g( r; r ') ( r ') u( r ') V ε, (5) is the Green s unction in the reerence (background) medium, and the integral is over the whole volume o medium. his is the Lippmann-Schwinger integral equation. Since the ield u( r ) equation (5) is not an explicit solution but an integral equation. under the integral is the total ield which is unknown,.1.1. Born approximation Approximating the total ield under the integral with the incident ield u ( ) Born Approximation r, we obtain the u r = u r + k d r g r r r u r 3 ( ) ( ) ' ( ; ') ε ( ') ( '). (6) V In general, the Born approximation is a weak scattering approximation, which is only valid when the scattered ield is much smaller than the incident ield. his implies that the heterogeneities are weak and the propagation distance is short. However, the valid regions o the Born approximation are very dierent or orward scattering than or backscattering. Forward scattering divergence or catastrophe is the weakest point o Born approximation. For simplicity, we use orescattering to stand or orward scattering. As can be seen rom equation (6), the total scattering ield is the sum o scattered ields rom all parts o the scattering volume. Each contribution is independent rom other contributions since the incident ield is not updated by the scattering process. In the orward direction, the scattered ields rom each part propagate with the same speed as the incident ield, so they will be 7

8 coherently superposed, leading to the linear increase o the total ield. he Born approximation does not obey energy conservation. he energy increase will be the astest in the orward direction, resulting in a catastrophic divergence or long distance propagation. On the contrary, backscattering behaves quite dierently rom orescattering. Since there is no incident wave in the backward direction, the total observed ield is the sum o all the backscattered ields rom all the scatterers. However, the size o coherent stacking or backscattered waves is about λ/4 because o the two-way travel time dierence. Beyond this coherent region, all other contributions will be cancelled out. For this reason, backscattering does not have the catastrophic divergence even when the Born approximation is used. his can be urther explained with the spectral responses o heterogeneities to scatterings with dierent scattering angles. From the analysis o scattering characteristics, we know that the orescattering is controlled by the d. c. component o the medium spectrum W ( ), but the backscattering is determined by the spectral component at spatial requency k, i.e. W ( k ), where k is the background wavenumber (see, Wu and Aki, 1985; Wu, 1989). he d. c. component o the medium spectrum is linearly increasing along the propagation distance, while the contribution rom W( k) is usually much smaller and increases much slower than W(). he validity condition or the Born approximation is the smallness or the scattered ield compared with the incident ield. hereore, the region o validity or the Born approximation or backscattering is much larger than that or orescattering. he other dierence between backscattering and orescattering is their responses to dierent types o heterogeneities. he backscattering is sensitive to the impedance type o heterogeneities, while orescattering 8

9 mainly responds to velocity type o heterogeneities. Velocity perturbation will produce travel-time or phase change, which can accumulate to quite large values, causing the breakdown o the Born approximation. his kind o phase-change accumulation can be easily handled by the Rytov transormation. his is why the Rytov approximation perorms better than the Born approximation or orescattering and has been widely used or long range propagation in the case o orescattering or small-angle scattering dominance..1.. Rytov approximation Let u ( ) r be the solution in the absence o perturbations, i.e., ( + ) u = k, (7) and the perturbed wave ield ater interaction with the heterogeneity as u( r ). We normalize u( r ) by the unperturbed ield u ( ) phase perturbation unction ψ ( r ), i.e. r and express the perturbation o the ield by a complex u ψ ( r ) ( r ) u ( r ) = e. (8) his is the Rytov ransormation (see atarskii, 1971; or Ishimaru, 1978, p.349). ψ ( r ) denotes the phase and log-amplitude deviations rom the incident ield: ψ = u = i ( φ φ ) A logu log log +, (9) A where φ and φ are phases o perturbed and unperturbed waves. Combining (), (7) and (8) yields ( ψ ψ ε ) u ψ + u ψ = u + k. (1) he simple identity together with (7) results in ( u ψ ) = ψ u + u ψ + u ψ, 9

10 From (1) and (11) we obtain ( k )( ψ ) u ψ + u ψ = + u. (11) ( + k )( u ψ ) = u ( ψ ψ + k ε ). (1) he solution o (1) can be expressed as an integral equation: 3 u ( r ) ψ ( r ) = d r ' g( r; r ') u ( r ')[ ψ ( r ') ψ ( r ') + k ε V ( r ')], (13) where g( r;r' ) the incident ield at is the Green s unction or the background medium, u ( r ') and u ( r ) r ' and r, respectively. are Equation (13) is a nonlinear (Ricatti) equation. Assuming ψ ψ is small with respect to k ε, we can neglect the term ψ ψ and obtain a solution known as the Rytov approximation: k 3 ψ ( r ) = d r ' g( r; r ') ( r ') u ( r ') ε (14) u V ( r ) Now, we discuss the relationship between the Rytov and Born approximations, and their strong and weak points, respectively. By expanding e ψ into a power series, the scattered ield can be written as ψ 1 ( e 1) = u ψ + u ψ + u u = u. (15) When ψ << 1, i.e., the accumulated phase change is less than one radian (corresponding to about one sixth o the wave period), the terms o ψ and higher terms can be neglected, and u u 3 = u ψ = k d r ' g, (16) V ( r; r ') ε( r ') u ( r ') which is the Born approximation. his indicates that when ψ << 1, the Rytov approximation reduces to the Born approximation. In the case o large phase-change accumulation, the Born approximation is no longer valid. he Rytov approximation still holds as long as the condition ψ ψ << k ε is satisied. 1

11 Let us look at the implication o the condition ψ ψ << k ε or the Rytov approximation. Assume that the observed total ield ater wave interacted with the heterogeneities is nearly a plane wave: ik1 r, u = Ae which could be the reracted wave in the orward direction, or the backscattered ield, where k ˆ 1 = kk1 and ˆk 1 is a unit vector. Since the incident wave is ik r u = A e, the complex phase ield ψ can be written as ψ = log AA + ik k r, (17) ( ) ( 1 ) and ψ = log A A + i k k ( ) ( 1 ) ψ ψ = log + log, (18) ( A A ) k k i( k k ) ( A A ) 1 1. (19) Normally wave amplitudes vary much slower than the phases, so the major contribution to ψ ψ in (19) is rom the phase term k. hereore, the condition or Rytov approximation can be approximately stated as 1 k k k k k 1 = 4 sin θ << ε, () where θ is the scattering angle. hereore the Rytov approximation is only valid when the scattering angle (delection angle) is small enough to satisy θ 1 sin << ε = 4 1 c c c ( r ) ( r ). (1) his is a point-to-point analysis o the contributions rom dierent terms (or example, the terms in the dierential equation (1)). For the integral equation (13), one needs to estimate the integral eects o ψ ψ and k ε. he heterogeneities need to be smooth enough to 11

12 guarantee the smallness o the integral o ψ ψ which is related to scattering angles, in comparison with the total scattering contribution k ε. In any case, the Rytov approximation is totally inappropriate or backscattering. In the exactly backward direction, θ = 18 and θ sin = 1, inequality (1) is hardly satisied. hereore, although not explicitly speciied, the Rytov approximation is a kind o small angle approximation. ogether with the parabolic approximation, they ormed a set o analytical tools widely used or the orward propagation and scattering problems, such as the line-o-sight propagation problem (e.g. Flatté et al., 1979; Ishimaru, 1978; atarskii, 1971). he Rytov approximation is also used in modeling transmission luctuation or seismic array data (Wu and Flatté, 199), diraction tomography (Devaney, 198, 1984; Wu and oksöz, 1987). atarskii (1971, Chapter. 3B) has some discussions on the relation between the Rytov approximation and the parabolic approximation... De Wol approximation We see the limitations o both the Born and Rytov approximations. Even in weakly inhomogeneous media, we need better tools or wave modeling and imaging or long distance propagation. Higher order terms o the Born series (deined later in this section) may help in some cases. However, or strong scattering media, Born series will either converge very slowly, or become divergent. hat is because the Born series is a global interaction series, each term in the series is global in nature. he irst term o the Born approximation is a global response, and the higher terms are just global corrections. I the irst term has a big error, it will be hard to correct it with higher order terms. One solution to the divergence o the scattering series is the renormalization procedure. Renormalization methods try to split 1

13 the operations so that the scattering series can be reordered into many sub-series. We hope that some sub-series can be summed up theoretically so that the divergent elements o the series can be removed. he De Wol approximation splits the scattering potential into orescattering and backscattering parts and renormalizes the incident ield and Green s unction into the orward propagated ield and orward propagated Green s unction (orward propagator), respectively (De Wol, 1971, 1985). he orward propagated ield u is the sum o an ininite sub-series including all the multiple orescattered ields. he orward propagator G is the sum o a similar sub-series including multiple orescattering corrections to the Green s unction. For backscattering, only the single backscattered ield is calculated at each step, and then propagated in the backward direction using the renormalized orward propagator (Green s unction) G. he De Wol approximation is also called the one-return approximation (Wu, 1996, 3; Wu and Huang, 1995; Wu et al., a, b), since it is a multiple-orescattering-single-backscattering (MFSB) approximation. It is also a kind o local Born approximation since the Born approximation applies only locally to the individual thin-slabs. From previous sections we know that the Born approximation works well or backscattering. With the renormalized incident ield and Green s unction, the local Born (MFSB) proved to work surprisingly well or many practical applications. he key is to have good orward propagators. Rino (1988) has obtained better approximation than MFSB in the wavenumber domain and pointed out the error o the De Wol approximation in the calculation o backscattering enhancement. he error (overestimate) is again due to the violation o the energy conservation law by the Born approximation. Even with a orescattering correction, the backscattered energy is still not removed rom the orward 13

14 propagated waves or local Born approximation. However, or short propagation distances in exploration seismology and some other applications, the errors in relection amplitudes may not become a serious problem. In the ollowing, we will adopt an intuitive approach o derivation to see the physical meaning o the approximation. he De Wol approximation bears some similarity to the wersky approximation in the case o discrete scatterers (wersky, 1964; Ishimaru, 1978). he wersky approximation includes all the multiple scattering, except the reverberations between pairs o scatterers that excludes the paths which connect the two neighboring scatterers more than once. he wersky approximation has less restrictions and thereore a wider range o applications than the De Wol approximation. he latter needs to deine the split o orward and back scatterings. We deine the scattering to the orward hemisphere as orescattering and its complement as backscattering. he Lippmann-Schwinger equation (5) can be written symbolically as u = u + G ε u, () where ε is a diagonal operator in space domain, and G is a nondiagonal integral operator. I the reerence medium is homogeneous, G will be the volume integral with the Green s unction g ( r; ') r as the kernel. Formally, () can be expanded into ininite scattering series (Born series) u = u + G ε u + G ε G ε u +. (3) I we split the scattering potential into the orescattering and backscattering parts ε = ε + (4) ε b and substitute it into (3), we can have all combinations o multiple orescattering and 14

15 backscattering. We neglect the multiple backscattering (reverberations), i.e., drop all the terms containing two or more backscattering potentials, resulting in a multiple scattering series which contains terms with only one he general term will look like ε b. G ε Gε Gε bgε Gε u. (5) he multiple orescattering on the let side o ε b can be written as G m m [ G ε ] G =, (6) and on its right side, u n n [ G ε ] u =. (7) Collecting all the terms o m G and u n respectively, we have M m= n= m ε M G = G G, N N u = Gε u n. (8) Let M and N go to ininite, then the renormalized G (orward propagator) and u (orescattering corrected incident ield) are: m= n= m ε G = G G, u = Gε u n, (9) and the De Wol approximation (in operator orm) becomes u = u + G ε u. (3) b he observed total ield u in (3) is dierent or dierent observation geometries. For transmission problems, the backscattering potential does not have any eect under the De Wol approximation, u transmissi on = u. (31) 15

16 On the other hand, or relection measurement, that is, when the observations are in the same level as or behind the source with respect to the propagation direction, there is no u in the total ield, Write it into integral orm, (3) becomes u r = u r + u relection = G ε bu. (3) ( ) ( ) d r ' g ( r, r ') b ( r ') u ( r ') V 3 ε. (33) Note that both the incident ield and the Green s unction have been renormalized by the multiple orescattering process through the multiple interactions with the orward-scattering potentialε..3. he De Wol Series (DWS) o multiple scattering De Wol approximation can be considered as the irst term o a multiple scattering series: the De Wol series. Ater substituting the decomposition (4) into the Born series (3), we rearrange and recompose the scattering series into a series according to the power o the backscattering potentialε b. he irst order in εb will be the De Wol approximation (one-return approximation). he second order corresponds to the double backscattering (double relection, double return) term. he higher terms represent the multiple backscattering series. he whole multiple scattering series (3) can be reorganized into u = u + G ε u + G ε G ε u + M m = [ Gεb] u. m= b b b (34) We see that the zero order term is the orward scattering approximated direct waveield. It is the direct transmitted wave in the real media. he irst order term is the De Wol approximation, which corresponds to the single backscattering signal, or the primary relections, as called in exploration seismology. his single backscattering signal is dierent 16

17 rom the Born approximation where the incident ield and the Green s unction are both deined in the background medium (a homogeneous medium). he De Wol series and generalized Bremmer series. Here we point out the dierences between the De Wol series (DWS) and the Generalized Bremmer Series (GBS).he original Bremmer series (Bremmer, 1951) is a geometric-optical series or stratiied media, which can be considered as a higher order extension to the regular WKBJ solution (the irst order term). Later it was generalized to 3D inhomogeneous media, and was named the generalized Bremmer series (Corones, 1975; De Hoop, 1996; Wapenaar, 1996, 1998; van Stralen et al., 1998; homson, 1999; Le Rousseau and de Hoop, 1). he zero order term (the leading term) o the GBS is a high-requency asymptotic solution (a WKBJ-like solution or Rytov-like solution) (de Hoop, 1996), and used as the Green s unction or deriving the higher order terms. he Green s unction is not updated when calculating the higher order scattering. hereore, it is similar to the Born series in a certain sense. Unlike the DWS, which is a series in terms o medium velocity variation, the GBS is in terms o the spatial derivatives o the medium properties (de Hoop, 1996). Because o the asymptotic nature o its Green s unction, the media need to be smooth on the scale o the irradiating pulse (De Hoop, 1996, Van Stralen et al., 1998; homson, 1999). Some authors used an equivalent medium averaging process to smooth the medium beore the application o the method (Stralen et al., 1998). Wapenaar s approach (1996, 1998) is to get an asymptotic solution, without averaging the medium, using the lux normalized decomposition o waveield. homson (1999, 5) included the second term o the asymptotic series to the asymptotic Green s unction to improve the amplitude accuracy. his zero-order term 17

18 solution or generally inhomogeneous media can be useul or seismic imaging and inversion (e.g. Berkhout, 198, Berkhout and Wapenaar, 1989). With careul amplitude correction, this type o Green s unction is an energy-lux conserved propagator in any heterogeneous media (including discontinuities), and is called (Wu et al., 4; Wu and Cao, 5) the transparent propagator. he irst term in the GBS (de Hoop, 1996; Wapenaar, 1996) in act is not the primary relection rom the real media, but a primary relection on the basis o the asymptotic Green s unction. Because the incident ield and Green s unction are not updated in deriving the irst order term, the primary relection thus obtained is similar to a distorted primary relection, ollowing the term o distorted Born approximation in the physics literature. he real primary relection, which is the irst term in the De Wol series, includes some multiple scattering terms in the GBS. 3. A DUAL-DOMAIN HIN-SLAB FORMULAION FOR ONE-REURN (MFSB) SYNHEICS he physical meaning o the one-return approximation is shown in Figure 1. First a preerred direction needs to be selected or the orward/backward scattering decomposition. In Figure 1 we choose the z-direction as the preerred one. We see that all the solid wave paths have only one backscattering with respect to the z-direction, and thereore will be included in the simulation using the one-return approximation. On the other hand, the dashed line has three backscattering points and cannot be modeled by the one-return approximation. Figure illustrates schematically the numerical implementation o one-return approximation using a thin-slab marching algorithm. he heterogeneous medium is sliced into numerous thin-slabs. Within each thin-slab, the local Born approximation can be utilized or the 18

19 calculation o the orward and backward scatterings. he orescattered ield is used to update the incident ield (the orward propagated waveield u ) and the backscattered ield u b is stored or later use. his procedure is iteratively done slab by slab, until the end o the model is reached. At the bottom o the model, u will be the primary transmitted wave. o get the primary relected waves, the marching will be done in the reversed direction, i.e. rom the bottom to the top o the model. At each thin-slab, the stored backscattered waveield will be picked up and propagated to the receiving point with the orescattering updated Green s unction G. In the next ew sections, the derivation and ormulations o the thin-slab and u screen approximations or scalar, acoustic and elastic waves will be given he case o scalar media Based on the De Wol approximation (33) and the marching algorithm shown in Figure, at each step we need to calculate the ore- and backscattered wave ields caused by the thin-slab. We can choose the thickness o the thin-slab to be thin enough so that the local Born approximation can be applied to the calculation. Replacing u in the Born approximation (6) with the updated local incident ield u, the scalar pressure ield at an observation point x * can be expressed as where 3 ( *) ( *) ( *; ) ( ) ( ) p x = p x + k d xg x x F x p x (35) V p is the local incident pressure ield and g, the Green's unction in the thin-slab. F(x) is the perturbation unction (scattering potential), F ( x) s = = c ( x) s c 1 ( x) s, (36) with s = 1 c as the slowness, where c is the velocity. he second term in the right hand side o (35) is the scattered ield and the volume integration is over the thin-slab. Choosing the 19

20 * * z-direction as the main propagation direction, the scattered ield at a receiving point ( x,z ) can be calculated as * (, * 3 * ) (, * ; = ) ( ) ( ) P x z k d xg x z x F x p x, (37) V where * x is the horizontal position in the receiver plane at depth z *. In the derivations o the next ew sections, we use a thin-slab geometry as illustrated in Figure 3. Within the slab the Green s unction is assumed to be a constant medium Green s unction g. Set z and z 1 as the slab entrance (top) and exit (bottom) respectively, and Fourier-transorm equation (37) with respect to x, resulting in where ( * ) z1 ( * = z ) P Κ, z k dz d x g K, z ; x F( x) p ( x ) (38) (, ;, ) * * i iγ z z ik x g K z x z = e e (39) γ is the wavenumber-domain Green s unction in constant media (see Berkhout, 1987; Wu, 1994), and γ = (4) k K is the vertical wavenumber (or the propagating wavenumber). Substituting (39) into (38) yields * * i z iγ z z 1 ik x (, ) = (, ) (, ) P K z k dze d x e F x z p x z. (41) z γ Note that the two dimensional inner integral is a -D Fourier transorm. hereore, the dual-domain technique can be used to implement (41). I x* = x 1, (41) is used to calculate the orward scattered ield and update the incident ield (transmitted waves) (35); On the other hand, i x* = x ', (41) is or the calculation o backscattered waves. In the case that only orward propagation is concerned, the iterative implementation o (35) and (41) composes a

21 one-way propagator. he derivation o (35) and (41) is based on the local Born approximation, however, the implementation in dual domains is similar to the classical phase-screen approach o a one-way propagator. It is known that the Born approximation is basically a low-requency approximation, and has severe phase errors or strong contrast and high-requencies. In order to have better phase accuracy, which is important or imaging (migration), certain high-requency asymptotic phase-matching has been applied to the local Born solution such that the travel time o the solution match exactly the geometric optical (ray) travel time in the orward direction. Even a zero-order matching leads to a solution better than the classic phase-screen method, i.e. the spit-step Fourier method (e.g. Stoa et al, 199). he method was originally called the "pseudo-screen" method (Wu and De Hoop, 1996; Huang and Wu 1996; Jin et al., 1998, 1999) to distinguish the new orm o screen propagator rom the classic phase-screen propagator. he phase-screen has operations only in the space domain so that the phase correction is accurate only or small-angle waves; while the pseudo-screen has operations in both the space and wavenumber domains to improve the accuracy or large-angle waves. he asymptotic phase-matching method used by Jin and Wu (1999) and Jin et al. (1998, 1999, ) in the hybrid pseudo-screen propagator applies a wavenumber ilter in the orm o continued raction expansion and can improve the large-angle wave response signiicantly. In the method, the wide-angle correction is implemented with an implicit inite dierence scheme and the expansion coeicients are optimized by phase-matching. he pseudo-screen propagator belongs to a more general category o generalized screen propagators (GSP) (Wu, 1994, 1996; Xie and Wu, 1998; de Hoop et al., ; Le Rousseau et al., 1). he hybrid GSP with inite dierence 1

22 wide-angle correction is similar to the Fourier-inite dierence (FFD) method (Ristow and Ruhl, 1994; Huang and Fehler, ). 3.. he case o acoustic media For a linear isotropic acoustic medium, the wave equation in requency domain is 1 ω p + p = ρ κ, (4) where p is the pressure ield, ρ and κ are the density and bulk module o the medium, respectively. Assuming ρ and κ as the parameters o the background medium, equation (4) can be written as 1 ρ ω p + κ p = ω p + p κ κ ρ ρ, (43) or ( + k ) p( x) = k F( x) p( x) which is the same as the case o scalar media except, (44) F 1 = + δ ρ k, ( x) δκ ( x) where F ( x) is an operator instead o a scalar unction, with (45) δ κ ( x) = κ κ 1 ( x), (46) and δ ρ ( x) = ρ ρ 1 ( x). (47) I ρ is kept constant ( ρ = ρ ), then δ = c, going back to the scalar medium κ c 1 case. Following derivation o equation (41), using the thin-slab geometry and the

23 dual-domain expression or the acoustic media, the scattered pressure ield at the receiving depth where * z can be written explicitly as { * * i z iγ z z 1 iκ x ( Κ, ) = δ (, ) (, z x κ x x ) P z k dze d e z p z γ i ˆ iκ x + k d xe δ ρ ( x, z) p ( x, z), k (48) and 1 k ˆ = ( Κ, kz), (49) k k = ± γ or orescattering and backscattering, respectively. he incident ield z p ( x, z) and its gradient p ( x, z) (, z ) p x at the slab entrance and at depth z can be calculated rom the ield 1 p = p ( ) z = d p e e 4π iγ z z i ( ) (, ) ( ) Κ x x x Κ Κ, (5) where ik ( ) ( ) ˆ iγ z z iκ x p x d Κ kp ( K ) e e, (51) 4π = 1 kˆ = ( Κ, γ ). (5) k Numerical tests or relected waves in acoustic media In the dual-domain thin-slab ormulation, no small-angle approximation is made. he only approximation is the smallness o perturbations within each thin-slab so that the background Green's unction can be applied and the incident waves can be treated as propagated in the background media. In the limiting case, the thickness o the thin-slab can be shrunk to a one grid step. In that case, the only approximation involved is the dual-domain implementation (or split step algorithm). Numerical tests on the wide-angle version (thin-slab 3

24 approximation) o the acoustic one-return method showed that the relection coeicients calculated rom the synthetic acoustic records agree well with the theoretical predictions when the incidence angles are smaller than the critical angle in the case o high-velocity layer relection, and to approximately 7 in the case o low-velocity layer relection (Wu and Huang, 199, 1995; Wu, 1996) he case o elastic media he equation o motion in a linear, heterogeneous elastic medium can be written as (Aki and Richards, 198) ω ρ( x) u( x) = σ ( x), (53) where u is the displacement vector, σ is the stress tensor (dyadic) and ρ is the density o the medium. Here we assume no body orce exists in the medium. We know the stress-displacement relation 1 σ ( x) = c( x) : ε ( x) = c : ( u + u ), (54) where c is the elastic constant tensor o the medium, ε is the strain ield, u stands or the transpose o u, and : stands or double scalar product o tensors deined through (ab) : ( cd) = ( b c)( a d ). Equation (53) can be written as a wave equation o the displacement ield: ωρ( xux ) ( ) = c: ( u+ u ). (55) 1 I the parameters o the elastic medium and the total wave ield can be decomposed as ρ( x) = ρ + δρ( x ), cx ( ) = c +δcx ( ), ux ( ) = u( x) + Ux, ( ) (56) 4

25 where ρ andc are the parameters o the background medium,δρ andδc are the corresponding perturbations, u is the incident ield and U is the scattered ield, then (55) can be rewritten as: 1 ωρu c : ( U+ U ) = F (57) where [ : ] F= u+ c ω δρ δ ε is the equivalent body orce due to scattering. Similar to (35), we can write the equation o the De Wol approximation or elastic displacement ield as { δρω δ ε } ( ) 3 (, ) = (, ) + (, ) + : (, ), ;, V ux z u x z d x u x z c x z G x z x z. (58) Following the derivation o equation (38), we can express the scattered displacement ield or a thin-slab in the horizontal wavenumber domain using local Born approximation as where { δρω δ ε } ( ) z 1 (, ) = (, ) + : (,, ;, z UK z dz d x u x z c x z G K z x z (59) ( ) ˆ ˆ ik ikα ik, ;, ( ˆ ˆ ) α r β = α α + β β 1 1 z z k k e k k e β ρω γ ρω γ ik r G K x I, (6) where I is the unit dyadic, and α β γ = k K, α α γ = k K, (61) β β where k α = ω / α and k β = ω / β are P and S wavenumbers with α and β as the P and S wave background velocities o the thin-slab, respectively. For isotropic media, δcx ( ): ε( x) = δλ( x) ε I+ δµ ( x) ε( x. ) (6) 5

26 Substituting (6) into (59), we can derive the dual-domain expressions or scattered displacement ields in isotropic elastic media. For P to P scattering: U PP ik z1 α α ikz ( z z) K z dze = γ z (, ) α ˆ ˆ ik x kk α α dxe uα( x, z) ˆ ik x kα d xe uα( x, z) λ + µ ikα ( ) δρ( x, z) ρ δλ( x, z) 1 δµ ( x, z) 1 + ˆ ˆ ˆ ik x kα kαkα : d xe εα ( x, z), λ µ ikα (63) with α α k z = +γ α or orescattering and z = γ α k or backscattering, and ˆ α k = ( K, k ) k. α z α Note that we replaced ρ, λ, µ in denominators by ρ = ρ + δρ, λ = λ + δλ and µ = µ + δµ. his replacement is the result o asymptotic matching between the Born approximation or large-angle scattering and the h- asymptotic travel-time (phase) or orward propagation. It is proved (Wu and Wu, 3) that with this replacement (asymptotic matching), the phase-shit in the exact orward direction is accurate and the phase error or small angles is reduced compared with the Born approximation. In the meanwhile, the phase error or large angle scattering is much smaller than that o the phase screen approximation. In (63) u α ( x, z), ( x, z) and ε ( x, z) can be calculated by: u α α 1 uα( x, ) = K α( K ) 4π ' ' ik x iγ ( z z ) z d e u e α 1 ' ' 1 ik x ˆ' iγ ( z z ) (, z) d e k ( ) e α uα x = α α, ik 4π K u K α 1 ' 1 ' ik 1 x ˆ ' ' ˆ ' ' εα ( x, z) = d Ke [ kαuα( K) + kαuα( K)] e ik 4π α 1 ' ' ' ' ik x ˆ' ˆ ' ' iγα ( z z ) d Ke k k u ( K) e, = 4π α α α ' ' iγα ( z z ) (64) ' ' where uα ( K ) = uα ( K ) and k ˆ α = ( K, γ α) kα. 6

27 For P to S scattering: PS ik * * β z β 1 ik ( ) ˆ ˆ (, ) z z z ik x δρ x z = ' z β β α U ( K, z ) dze {( I k k ) d x e u ( x, z) γ ρ β ˆ ˆ ˆ δµ ( x, z) 1 ( I k k ) [ k d x e ( x, z)]}, ik x β β β εα µ ikβ (65) where ˆ β k = ( K, k ) k. β For S to P scattering: z β SP * * ik z α α 1 ik ( ) ˆ ˆ (, ) z z z ik x δρ x z = ' z α α β U ( K, z ) dze { k k d x e u ( x, z) γ ρ α kα ˆ ˆ ˆ δµ ( x, z) 1 ( ) k ( k k ): d e ( x, z)}. k β ik x α α α x ε β µ ikβ (66) For S to S scattering: SS * * ikβ z β 1 ik ( ) ˆ ˆ (, ) z z z ik x δρ x z = ' z β β β U ( K, z ) dze {( I k k ) d x e u ( x, z) γ ρ β ˆ ˆ ˆ δµ ( x, z) 1 ( I k k ) [ k d x e ( x, z)]}. ik x β β β ε β µ ikβ (67) In equations (66) and (67), ( z u x ) and ε ( z β x, ) can be calculated by where ˆ = ( K, γ ) k. β, 1 u ( z) d e ( ) e β β x, = u K 4π ik i ( z z ) x γ K β ik ˆ ˆ ( ) x iγ z z ( z) d [ ( ) ( ) ], e k k e β ε β x, = β β + β β ik 4π K u K u K (68) β k β β β 3.4. Implementation procedure o the one-return simulation Under the MFSB approximation we can update the total ield with a marching algorithm in the orward direction. We can slice the whole medium into thin-slabs perpendicular to the propagation direction. Weak scattering condition holds or each thin-slab. For each step orward, the orward and backward scattered ields by a thin-slab between z and z 1 are calculated. he orescattered ield is added to the incident ield so that the updated ield 7

28 becomes the incident ield or the next thin-slab. he procedure or acoustic and elastic media can be summarized as ollows (see Figure ). he simpliication or the case o scalar media is straightorward. 1. Fourier transorm (F) the incident ields into wavenumber domain at the entrance o each thin-slab.. Free propagate in wavenumber domain and calculate the primary ields and its gradients (including strain ields or the case o elastic media) within the slab. 3. Inverse F these primary ields and its gradients into space domain, and then interact with the medium perturbations: calculation o the distorted ields. 4. F the distorted ields into wavenumber domain and perorm the divergence (and curl, in the case o elastic media) operations to get the backscattered ields. Sum up the scattered ields by all perturbation parameters and multiply it with a weighting actor i γ, then propagate back to the entrance o the slab. 5. Calculate the orescattered ield at the slab exit and add to the primary ield to orm the total ield as the incident ield at the entrance o the next thin-slab. 6. Continue the procedure iteratively until the bottom o the model is reached. 7. Propagate the backscattered waves rom the bottom up to the surace and sum up the contributions o all the thin-slabs during the propagation. Note that medium-wave interaction, or the case o acoustic waves, involves vector operations and needs 3 pairs o ast Fourier transorms (FF s) or each step, while or the case o elastic waves, tensor (strain ields) operations are involved. Due to the symmetric properties o the strain tensors, there are only 6 independent components or 8

29 each tensor. From (63) - (68) we see that many pairs o FF s are required or each step and thereore the computation or elastic wave scattering is rather intensive. 4. FAS ALGORIHM OF HE ELASIC HIN-SLAB PROPAGAOR AND SOME PRACICAL ISSUES 4.1. Fast Implementation in dual domains From equations (63) to (68), we see that the leading-order interactions between incident ields and heterogeneities are expressed in three-dimensional volume integrals. Also the scattered and incident wavenumbers are coupled with each other. So the computation o these equations is still intensive. In this section, the parts o the integration over z in the equations are analytically estimated. Assume that the slab or each marching step is thin enough that the parameters (velocity and density) can be approximately taken as invariant along z, the integration with respect to z in equation (63) can be calculated as z1 z dze α ikz ( z z) + iγ α ( z z ) i( γα + γ α ) z/ γα γ α ze sinc z or orescattering ( z z1), = = i( γα + γ α ) z/ γα + γ α ze sinc z or backscattering ( z = z ). (69) We see that the integration over z has been done analytically; however, γ α and γ α are still coupled, which prevents the ast computation o the thin-slab method. o decouple γ α and γ α, we neglect the angular variation o amplitude actors but keep the phase inormation untouched by taking the approximation γ = = k or the amplitude actors in equation α γ α α (69). his assumption is valid or the case where the small-angle scattering is dominant, and thereore the direction o the scattered waves are not ar rom the incident direction. Under this approximation, equation (69) becomes 9

30 z1 z dze α ikz ( z z) + iγ α ( z z ) i( γα γ α ) z ze + / or orescattering ( z = z1), i( γα+ γ α ) z/ ze sinc( kα z) or backscattering ( z = z ). (7) For the scattered ields P-S, S-P and S-S, similar approximations can be obtained as ollows. For P-S or S-P scattering, / = i( γ α + γβ ) z/ ze sinc[( kα kβ) z ] or orescattering ( z z1), z1 β ikz ( z z) iγ α ( z z ) dze + (71) z i( γ α + γβ ) z/ ze sinc[( kα kβ) z ] or backscattering ( z z ). For S-S scattering, + / = i( γβ γ ) z β ze + / z = z1 or orescattering ( ), z1 β ikz ( z z) iγ β ( z z ) dze + (7) z i( γβ + γ β ) z/ ze sinc( k β z) or backscattering ( z = z ). Ater integration over z, the integration over transverse plane x in equations (63) (68) can be carried out by the FF. In order to urther expedite the computation, we can group the P PP SP scattered ield equations (63) to (68) into U ( K, z ) = U ( K, z ) + U ( K, z ) U S PS SS ( K, z ) = U ( K, z ) + U ( K, z ), i.e., and P ikα iγ z i ( ) PP SP ( ) ˆ ˆ z e α / K x δρ x U K, = zk k d e η α( ) + η β( ) α x α γα ρ u x u x ik ( ) 1 x δλ x PP d xe η α ( ) λ + µ ik u x α ik ( ) 1 x δµ x PP SP ( kˆ kˆ ): d xe η εα ( ) + η εβ ( ), α α λ + µ ik x x (73) α ik U ( K ) (I ) x u ( x ) u ( x ) S β iγ z i ( ) PS SS ˆ ˆ z e β / δρ, = z k k d e η β β α + η β γβ ρ K x x i δµ ( ) 1 PS SS kˆ d e η εα ( ) η εβ ( ) K x x x + β µ ik x x, (74) β where z ( z = z or z = z ) indicates the position o the receiver plane. he modulation 1 actors PP η, η SP PS = η and SS η are 3

31 η η η PP PS SS 1 or orescattering =, sinc( kα z) or backscattering sinc[( kα kβ) z/ ] or orescattering =, sinc[( kα + kβ) z/ ] or backscattering 1 or orescattering =. sinc( kβ z) or backscattering (75) (76) (77) Note that the actors iγ ( z z ) e α and iγ ( z z ) e β have been replaced by iγ z e α / and iγ z e β / or calculating the background ields. he phase matching (asymptotic matching) has been applied in equations (73) and (74). Under the above approximations, the thin-slab ormulas may be implemented by the procedures as used in the complex screen method (Wu, 1994; Xie and Wu, 1). First, the whole medium is sliced into appropriate thin-slabs along the overall propagation direction. Weak scattering conditions hold or each thin-slab and the parameters can be considered invariant within each thin-slab in the preerred propagation direction. Suppose that all incident ields, at the entrance o each thin-slab, are given in the wavenumber domain. he implementation procedures may be summarized as ollows: 1. Freely propagate in the wavenumber domain and calculate the primary ields, the divergence o incident P wave, and strains (equations 64 and 68).. Inverse-FF the primary ields, the divergence and strains into the space domain, and then calculate the distorted ields (the space domain unctions beore the F in equations (73)-(74)). 3. Calculate the orescattered ields at the thin-slab exit, and add the orescattered ields to the 31

32 primary ields to orm the total ields as the incident ields or the next thin-slab. In the same time backscattered ields at the thin-slab entrance are also calculated. 4. Continue the procedures 1 to 3 iteratively until the last thin-slab, and the total transmitted ields are obtained at the exit o the last thin-slab. 5. Propagate back the backscattered ields rom the last to the irst thin-slabs using a similar iterative procedure as step 1 to 3, and sum up all backscattered ields generated by each thin-slab to get the total relected ields at the surace. Let us estimate the computation speed. Most calculations involve only ast Fourier transorms. he total computation time can be estimated rom the time used in calling FFs. aking the complex screen method as a reerence, or a -D case, the thin-slab method needs 11 inverse FF s and 11 orward FF s, while the complex screen method needs 5 inverse FF s and 7 orward FF s. For a 3-D case, the thin-slab method needs 19 inverse FF s and 19 orward FF s, while the complex screen method needs 7 inverse FF s and 1 orward FF s. he thin-slab method takes about twice as much time as the complex screen method does, but is still much aster than the ull wave methods. We will see a comparison o computation time between the thin-slab method and inite dierence method in section Incorporation o boundary transmission/relection into the one-return method For the one-return elastic wave propagators mentioned above, the boundary transmission and relection or a thick-layer (much thicker than the wavelength) is ormed by the superposition and intererence o numerous thin-slab scatterings. his gives the lexibility o the thin-slab propagator to treat arbitrarily irregular interace. However, or a 3

33 homogeneous thick-layer, the calculation o boundary relection/transmission (R/) using the relectivity method (see Aki and Richards, Chapter 5, 198) is very eicient and accurate. hereore the relectivity method has been incorporated into the thin-slab propagator (Wu and Wu, 3). In addition to the increase o eiciency, this can also reduce the accumulated errors o the thin-slab propagator when propagating in a thick layer with strong contrast o parameters rom the surrounding medium. Since the thin-slab method is based on a perturbation approach, the choice o the background medium is an important issue. o make the perturbation small, the background medium parameters are changed as soon as the waves enter a laterally homogeneous region. ake the model shown on Figure 1 as an example. On top o the irregular structure (with grey color) is a homogenous background. When the wave enters the laterally heterogeneous part o the medium, the thin-slab propagator is used or the propagation. We can put in an artiicial boundary as shown in the model with bold lines. At that artiicial boundary, the reerence parameters will jump rom the top medium to the grey structure, a -1% jump. he relection/transmission at that boundary can be calculated using the analytical ormulation, such as the Zoeppritz equation, and then the propagation can be done, with an elastic propagator in homogeneous media, to the bottom by one big step. Note that the artiicial boundary is added only to acilitate the calculation. he relected ield generated by the relectivity calculation will be cancelled by the accumulated thin-slab scattered ield. hereore the artiicial boundary will not generate spurious arrivals. he thin-slab propagator is a dual-domain (space-wavenumber domains) approach. In a wavenumber domain, the waveield can be expressed by a superposition o plane waves. 33

34 hereore, incorporating a relectivity calculation into the thin-slab propagator is straightorward. Ater plane wave decomposition, the spectra o incident ields can be decomposed into P- and SV- and SH-components, and the R/ ormulation can be applied directly to these components. In section 4.4 numerical examples will be given to show the validity and eiciency o this approach reatment o anelasticity: the Q-actor Spatially varying quality actors ( Q andq ) can also be incorporated into the thin-slab p s propagator to study the eects o intrinsic attenuation in visco-elastic media. Spatially varying intrinsic attenuation can cause not only waveield attenuation but also scattering and requency-dependent relections. he thin-slab propagator, with spatially varying Q-actor, is an eicient tool or such study. For elastic, isotropic media, Lamé constants λ, µ, λ, and µ are related with elastic parameters by λ = ρα ρβ, µ = ρβ, (78) λ = ρα ρβ, µ = ρβ (79), whereα, and β are compressional- and shear-wave velocities o background medium. We introduce complex velocities by perorming the ollowing transorms: where Q P and α α( 1 ), β β( 1 ), (8) i Q P i Q S α ( i Q P ) β β ( i Q S ) α 1, 1, (81) Q S are compressional- and shear-wave quality actors o background 34

35 medium. i is the imaginary unit. Once all parameters in equations (8)-(81) are known, we can calculate the perturbations δλ and δµ using equations (78)-(79). Now λ, and µ become complex. As a result, the reerence wavenumbers o P- and S-waves also become complex.he heterogeneities o quality actors (Q P and Q S ) have been included in the complex δλ and δµ. With the above extension, the dual-domain thin-slab propagators can be used to model visco-elastic seismic responses. In section 6.3 some numerical examples will be given Numerical tests or the elastic thin-slab method Relection coeicient calculations he ollowing examples show the angle dependence o amplitudes (relection coeicients) calculated by the thin-slab method. he model used is deined on a 48 rectangular grid. he grid spacing in the horizontal direction is 16 m and in the vertical direction is 4 m. A horizontal interace is introduced in the middle o the model. he upper 3 layer hasα = 36km. / s, β = 8km. / s and ρ = g/ cm, which is taken as background medium. he lower layer has dierent P and S wave velocities relative to the upper layer. A 15 Hz plane P wave (or S wave) is incident on the interace at dierent angles. o enhance the stability in the calculation o relection coeicients, we chose 5 samples (displacement amplitudes in the space domain) at the center o the model or both the incident and relected ields and calculate their means respectively. Relection/conversion coeicients are deined by the ratio o the relected amplitudes to the incident amplitudes at the same receiver plane. Figure 3 shows the relection/conversion coeicients versus angle o incidence with a perturbation o 1% or both P and S wave velocities. he theoretical relection/conversion 35

36 coeicients (dashed lines) are also given as reerences. he upper panel corresponds to P wave incidence and the lower panel, S wave incidence. he angles o incidence o S wave are limited to the critical angle o S-P converted wave. Both results agree well with the theory or a wide range o incident angles (55 o or P wave incidence, and near critical angle 3 o or S wave incidence). Figures 5 to 7 show similar results as shown in Figure 4 but with dierent perturbations o -1%, % and -%, respectively. Figure 5 corresponds to a negative velocity perturbation. For P wave incidence, no critical angle exists. However, errors occur or large angles o incidence (>65º or P wave incidence). his is limited by the ability o the one-way propagator to handle wide-angle waves. For S wave incidence, both results are in good agreement up to the critical angle. Figure 6 corresponds to a perturbation o % or both P and S wave velocities. For a small angle o incidence (4º or P wave incidence and º or S wave incidence), the thin-slab results match the theoretical values. Comparing Figures 6 and 4, we see that the wide-angle capacity o the thin-slab method decreases as perturbations increase. Figure 7 corresponds to a perturbation o -% or both P and S wave velocities D synthetic seismograms Synthetic seismograms are also generated to urther demonstrate the capability o the thin-slab method. Figure 8 shows a -D model that is a vertical slice cut rom the elastic French model (French, 1974). he model has a strong irregular interace that will generate large-angle relections and scattering. he parameters o the background medium are taken as α = 36km. / s, β = 8km. / s and 3 ρ = g. / cm. he layer in black color has a perturbation o -% or both P and S wave velocities. Source and receiver geometry are also 36

37 shown in the igure. A Ricker wavelet with a dominant requency o Hz is used. For the thin-slab method, the spacing interval is 8 m in both horizontal and vertical directions. For the inite dierence method, the spacing interval is 4 m and time interval is.5 ms. he stability criterion is satisied. he direct arrivals have been properly removed rom the inite dierence results. Figure 9 shows a comparison o the synthetic seismograms between the thin-slab method and inite dierence method. Both amplitudes and arrival times are in good agreement up to large osets (~1.4 km) compared to the depth (~1 km). For this example, the thin-slab method is about 57 times aster than the inite dierence method. he actor will increase as the size o model increases, especially or 3D cases D synthetic seismograms Figure 1 shows a 3D French model. he parameters o the background medium are taken as α = 3.6 km/s, β =.8 km/s and ρ =. g/cm 3. he Grey structure has a perturbation o -1% or both P- and S-wave velocities. A Ricker wavelet with a dominant requency o 1 Hz is used. For the 8th-order 3D elastic inite-dierence method (Yoon, 1996), the spacing interval is m. he actual grid size used is including 5 grids o absorbing boundary or each ace o the model. ime interval used is.1 sec and 5 time steps are calculated. It took about 8 hours. For thin-slab method, the spacing interval used is m in transversal plane, which is the same as used or the inite-dierence method. But a ine grid size o 5 m is used in propagation direction. We did the same calculation on the same machine using the thin-slab propagator. It took.7 hours. hin-slab is about 1 times aster than the inite-dierence method. Figure 11 gives the 3 components o the synthetic seismograms calculated using the inite-dierence method (solid) and by the 37

38 thin-slab method (dotted). he Y-component in Figure 11 has been multiplied by a actor o 3. We see that the results o the two methods are in excellent agreements or small to mild angle scatterings. Especially or the relected/converted waves generated at the lower interace o the model, the use o relectivity method improved the accuracy o the arrival times o those events, and matched well with those by the inite-dierence method. 5. HE SCREEN APPROXIMAION For some special applications, such as slowly varying media, the synthetics only involve small-angle scattering. In this case the screen approximation can be applied to accelerate the computation. Under small-angle scattering approximation, we can compress the thin-slab into an equivalent screen and thereore change the 3-D spectrum into a -D spectrum. Dual-domain implementation o the screen approximation will make the modeling o backscattering very eicient. In the ollowing, the cases o acoustic and elastic media will be given respectively Screen Propagators or acoustic media hin-slab ormulation in wavenumber domain In order to urther accelerate the computation, approximations to the interaction between the thin-slab and incident waves can be applied. First, we discuss the thin-slab ormulation in wavenumber domain, and in the next section, the screen approximation will be made based on the wavenumber domain ormulation. o obtain the wavenumber domain ormulation, we carry out analytically the integration along z-direction between the slab entrance z and the exit z 1 (see Figure 3). In the case o acoustic media, we substitute (5) and (51) into equation (48) and perorm the moving rame 38

39 coordinate transorm z z z, resulting in i k i( γ )( z z ) P( K, z ) = e ± dk γ 4π { z i( ± γ γ ) z xεκ (, x) ( K) dz d z p e e ˆ ˆ, i( K K ) x z i( ± γ γ ) z i( K K ) x ( k k ) dz d x ( ) ( ), ε ρ z x p K e e } (8) where p ( K ) = p ( z, K ) (83) is the incident ield at the slab entrance, z = z1 z, and ± γ correspond to orescattering and backscattering respectively. Note that z i( k k ) x x x = k k dz d F( ) e F ( ), (84) where F ( k) is the 3D Fourier transorm o the thin-slab, i.e. the Slab-Spectrum, k is the incident wavenumber (5), and k is the outgoing wavenumber (scattering wavenumber) deined as or orescattered ield and k = k = K +γ ˆ (85) ez b k = k = K γ ˆ (86) ez or backscattered ield. hereore the Local Born scattering in wavenumber domain can be written as k ikz ( z z ) P( K, z ) = i e 8πγ ˆ ˆ d εκ( kz γ ) ( k k ) ερ( kz γ ) K K K, K K, p ( K ), (87) with kz = γ or orescattering and k z = γ or backscattering. When the receiving level is at the bottom o the thin-slab (orescattering), z = z1 ; while z = z is or the backscattered 39

40 ield at the entrance o the thin-slab. he total transmitted ield at the slab exit can be calculated as the sum o the orescattered ield and the primary ield, which can be approximated as 1 iγ ( z z ) ik x p ( x, z ) = d ( ) p e e 4π K K. (88) We see that the scattering characteristics depend on the spectral properties o heterogeneities. In the case o large-scale heterogeneities, where lateral sizes o heterogeneities are much larger than wavelength, the major orescattered energy is concentrated in a small cone towards the orward direction. For orescattering, the outgoing wavenumbers k are nearly in the same direction as the incoming wavenumbers k. Within the small cone, k k stays small. hereore, the scattered waves are controlled by the low lateral spatial-requency components o heterogeneities. In the meanwhile, k and k have opposite directions or backscattered waves and the backscattering is most sensitive to those vertical spectral components which are comparable to the hal-wavelength (see Wu and Aki, 1985). Also, we know that orescattering is controlled by velocity heterogeneities, while backscattering responses mostly to impedance heterogeneities (ibid). his point will be seen much more clearly in the next section. Note that the wave-slab interaction in wavenumber domain [equation (87)] is not a convolution and thereore the operation in space domain is not local. hereore, the wave-slab interaction in wavenumber domain involves matrix multiplication and is computationally intensive Small-angle approximation and the screen propagators Under this approximation, both incoming and outgoing wavenumbers have small 4

41 transversal components K compared to the longitudinal component γ and thereore K γ = k K k(1 ). (89) k hen k k = ( K K ) + ( γ γ )ˆ e ( ) ( K K K K + )ˆ e k k ( K K ) + ˆ e z z z (9) and b k k = ( K K ) ( γ + γ ) eˆz ( K K ) eˆz. (91) k Screen approximation in acoustic media With the small-angle approximation, the 3D thin-slab spectrum (84) can be approximated by: F ( k k) F ( K K, K = ) z z i( K K ) x x ε κ x ερ x = d e dz[ (, z) (, z)] = i( K K ) x d xe SV( x), (9) where S V is a D screen o velocity perturbation, and b F ( k k) F ( K K, K = k) z z i( K K ) x ikz x ε κ x ερ x = d e dze (, z) + (, z) = i( K K ) x d xe SI( x), (93) where S I is a D screen o impedance perturbation. We see that with the above approximation, 3-D thin-slab spectra have been replaced by -D screen spectra that are slices o the 3-D spectra. In the case o orescattering, the slice is rom a velocity spectrum at K z =, where K z is the spatial requency in the z-direction; while or backscattering, rom an impedance spectrum, a slice at K = k. In the special case when F( x, z) varies z very little along z within the thin-slab, the screen spectra can be urther approximated as 41

42 F ( k k) ( K K ) S V = [ δ ( K K ) δ ( K K )], b F ( k k) S I( K K) κ ρ z = [ δ ( K ) ( )] sinc( ) ik z K + δ K K z k z e. (94) κ he scattered ields (74) under the screen approximation become ρ k ( ) ( ) ik z z z P K, z i e dk ( ) ( ) S K K p K. (95) 4πγ he above equation is a convolution integral in wavenumber domain and the corresponding operation in space domain is a local one. he dual-domain technique can be used to speed up the computation: ik ( ) ( ) ik z z z i K x P K, z e dxe S( x) p ( x ), (96) γ where 1 z S( x) = SV( x) = dz[ ( ) ( )] 4 δ κ x, z δρ x, z or orescattering, (97) 1 z ikz S( x) = SI( x) = dze [ ( ) ( )] 4 δ κ x, z + δρ x, z or backscattering. (98) he total transmitted ield at z 1 is 1 = ikz( z1 z ) ik X = e dxe p x + iksv x P ( K, z ) p ( K, z ) P ( K, z ) ikz( z1 z ) ik X x x V x = e d e p ( )exp iks ( ) ( ) 1 ( ), (99) where k 1 has been used or the scattered ield based on the small-angle scattering γ α approximation. he above equation is the dual-domain implementation o phase-screen propagation Procedure o MFSB using the screen approximation 1. Fourier transorm the incident ield at the starting plane into wavenumber domain and 4

43 propagate to the screen using a constant velocity propagator.. Inverse Fourier transorm the incident ield into space domain. Interact with the impedance screen (complex-screen) to get the backscattered ield and interact with the velocity screen (phase-screen) to get the transmitted ield. 3. Fourier transorm the transmitted ield into wavenumber domain and propagate to the next screen using a constant velocity propagator. 4. Repeat the propagation and interaction screen-by-screen to the bottom o the model space. 5. backpropagate and stack the stored backscattered ield screen by screen rom the bottom to the top to get the total backscattered ield on the surace. Similar to the derivation or acoustic media, the wavenumber domain ormulation can be obtained by substituting equations (64), (68) into equations (63) and (65)-(67). PP i P δρ( k ) δλ( k ) δµ ( k ) U ( K ) ˆ ˆ ˆ ˆ ˆ, K = ku α k ( k k ) ( k k ) α α, α α α γ α ρ λ + µ λ + µ PS i P δρ( k ) β δµ ( k ) U ( K ) ( ˆ ˆ ) ˆ ˆ, K = kβ I k k u ( k k ), β β α β γβ ρ α µ SP i S δρ( k ) β δµ ( k ) U ( K ) ( ˆ ˆ ˆ ˆ, K = kα u k ) k ( k k ) α α, β α γα ρ α µ (1) SS i S δρ( k ) δµ ( k ) U ( K ) ( ˆ ˆ ) ˆ ˆ, K = kβ I k k ( k k ) β β u β β γβ ρ µ S δµ ( k ) kˆ ( u ˆ β kβ), µ P where u is the spectral ield o the incident P wave, and δρ( k ), δλ( k ) and δµ ( k ) are the three-dimensional Fourier transorms o medium perturbations, and k = kˆ k ˆ is the 43

44 exchange wavenumber with ˆk and ˆk as incident and scattering wavenumber vectors, respectively. From the thin-slab ormulation, under the small-angle approximation, both incident and scattered wavenumbers have small lateral components K and K compared to vertical components and thereore γ α kα(1 K kα) γβ kβ(1 K kβ) /, /, γ (1 / ), (1 / ). α kα K kα γβ kβ K kβ (11) Using these approximations and neglecting higher-order (greater than second order) small quantities, the scattering patterns can be obtained as ( K K ) / kα or orescattering ( kˆ kˆ ) α α, 1 + ( K + K ) / kα or backscattering + 1 ( kβk kαk ) /( kαkβ) or orescattering ( kˆ kˆ ) α β, 1 + ( kβk + kαk ) / ( kαkβ) or backscattering + 1 ( kαk kβk ) /( kαkβ) or orescattering ( kˆ kˆ ) β α, 1 + ( kαk + kβk ) / ( kαkβ) or backscattering (1) ( K K ) / kβ or orescattering ( kˆ kˆ ) β β. 1 + ( K + K ) / kβ or backscattering o decouple the incident wavenumber K and the scattered wavenumber K in equations (1), suppose that the medium heterogeneities are smooth enough that the scattering wavenumbers ˆk α or ˆk β deviates not too ar rom the incident wavenumbers k ˆ ' α or k ˆ ' β. We can take K to be approximately equal to K, the wavenumber couplings may be simpliied urther by + 1 or orescattering ( kˆ ˆ ) α k α, 1 + K / or backscattering kα 44

45 + 1 ( K/ kβ K/ kα) / or orescattering ( kˆ kˆ ) α β, 1 + ( K / k + β K / kα) / or backscattering + 1 ( K/ kα K/ kβ) / or orescattering ( kˆ kˆ ) β α, 1 + ( K / k + α K / kβ) / or backscattering (13) + 1 or orescattering ( kˆ ˆ ) β k β. 1 + K / kβ or backscattering It can be seen rom equations (1)-(13) that the irst-order corrections have relatively stronger eects on the backscattering than on the orescattering. he scattering pattern ( I kˆ kˆ ) kˆ β β appearing in equation (13) is zero or orward scattering, while or β backward scattering it is ( I kˆ kˆ ) kˆ kˆ ( kˆ kˆ ) kˆ β β β = β β β β backward kˆ K k β + / backward 1 β kβ backward ˆ up to the irst-order correction. K / k, K / k, (14) β β 5.3. Complex screen method with irst-order corrections Numerical tests show that the eect o the irst-order corrections o wave couplings or P-S and S-P orescattering can be neglected. So the irst-order corrections are introduced only or backward propagators. Substituting equations (13)-(14) into equation (1) and integrating over incident wavenumber ˆk, we can obtain the ollowing ormulas U K K ˆ K δα( K ) η PP ' ' P PP (, ) = ikα zkαu ( ) α, (15) 45

46 ˆ ˆ ˆ ˆ δβ ( K ) β 1 δµ ( K ) U ( K, K ) = ik zu ( K )[ k k ( k k )][ + ( ) ] η PS ' P ' ' PS β α β α β β α µ ˆ ˆ δβ ( K ) β 1 δµ ( K ) U ( K, K ) = ik z( u ( K ) k ) k [ + ( ) ] η SP ' S SP α α α β α µ ˆ ˆ δβ ( K ) U ( K, K ) = ik z[ u ( K ) k ( u ( K ) k )] η SS ' S ' S ' SS β β β β PP i ik z PP K K α ( ˆ b z1) kαk ze η b ( 1 ) V V (1 ) V a ρ λ µ kα k α, (16), (17), (18) U K, = +, (19) U K, = I V + + V, (11) PS i i( k k ) z PS P 1 K K α + β / β P ( 1) ( ˆ ˆ b z kβ ze ηb k k ) ρ 1 β β µ α kα k β U K, = V + + V, (111) SP i i( k k ) z SP S 1 K K α + β / β S ( ˆ ˆ b z1) kα ze ηb k k ρ 1 α α µ α kα k β SS i ik z SS S K β S S ( 1) ( ˆ ˆ ) (1 ) ˆ b z kβ ze ηb kβkβ ρ µ K β kβ µ k β U K, = I V + V V, (11) where V ρ is the distorted P wave amplitude by density perturbation, P V ρ is the distorted P wave vector ield (displacement) by density perturbation, etc., deined as 46

47 ik ( ) x P δρ x Vρ ( K) = d xe u ( x, z ), ρ ik ( ) x P δλ x Vλ ( K) = d xe u ( x, z ), λ + µ ik ( ) x P δµ x Vµ ( K) = d xe u ( x, z ), λ + µ P ik ( ) x P δρ x Vρ ( K) = d xe u ( x, z ), ρ V K = d x e u x, z P ik x P µ ( ) ( ) β β β δµ ( x ), µ S ik ( ) x S δρ x Vρ ( K) = d xe u ( x, z ), ρ S ik ( ) x S δµ x Vµ ( K) = d xe u ( x, z ), µ K ( K ) = K / k, K / k, and the actors η, η, η, η, η, η and η SS, η are given by equations (75)-(77). PP PP b PS PS b SP SP b SS b In the above derivation, we assume that the thin-slab is thin enough so that the parameters can be approximated as unchanging along the z direction, and we also approximate γ α and γ β in the denominators o equations (1) by k α and k β, respectively. Although equations (19)-(11) look a little more complicated in orm than those in zero-order method, they can all be implemented using the FF and the eiciency is not compromised too much. It has been shown that the zero-order complex screen results agree with the thin-slab results only or small incidence angles. As incidence angle increases, the amplitude o the relected waves deviates gradually rom that by the thin-slab method. However, the irst order complex screen results are in good agreement with the thin-slab results or airly wide angles (4 o or P wave incidence and o or S wave incidence or the French model). 6. REFLECED WAVE FIELD MODELING USING HIN-SLAB MEHOD AVO analysis plays an important role in modern seismic data interpretation in 47

48 exploration seismology. AVO measures the angle-dependent relection response o an interace and relates the response to in-situ elastic parameters and luid contents o the target intervals. For lat interaces in homogeneous elastic media, AVO curves can be easily calculated theoretically. However, the observed relection responses o a seismic target are signiicantly aected by many other actors, such as data collection, data processing and wave propagation eects in heterogeneous media. Beore analyzing the AVO responses, these eects should be studied and compensated. Forward modeling can be useul in understanding wave propagation eects on AVO. In addition, orward modeling can also be useul in interpreting complicated AVO measurements, providing appropriate model parameters or data processing, and developing algorithms o requency-dependent AVO inversion (Dey-Sarkar and Svatek, 1993). here are several modeling algorithms available or AVO calculation. he relectivity method is one o the most common methods used or modeling AVO responses in layered media (Fuchs and Müller, 1971; Simmons and Backus, 1994; Wapenaar et al., 1999). It can generate an exact AVO response in arbitrarily layered medium, but cannot be used in structures with lateral velocity variations. Ray methods based on various approximations o the Zoeppritz equations are also very common or AVO analysis (Widess, 1973; Simmons and Backus, 1994; Bakke and Ursin, 1998). However, in the presence o thin layers, primaries-only Zoeppritz modeling may produce incorrect results. Another intrinsic limit o the ray methods is its inability in dealing with requency-dependent scattering associated with heterogeneities. A number o authors applied pseudospectral and inite-dierence methods to more complicated geologic models including anelasticity, overburden structure, scattering attenuation, and anisotropy 48

49 (Chang and McMechan, 1996; Adriansyah and McMechan, 1998; Yoon and McMechan,1996). In principle, these methods can deal with arbitrarily complicated geologic models. However, they are very time-consuming and memory demanding especially or 3-D models. his is also true or handling structures with thin layers where ine grids must be used. In this section we apply the elastic thin-slab method to AVO modeling in sedimentary rocks. Several numerical experiments, including relection coeicient calculations, relection synthetics with lateral parameter variations, and thin-bed AVO, have been conducted and compared with relectivity and inite-dierence methods. he accuracy and wide-angle capacity o the thin-slab method are demonstrated. Some examples showing the eects o lateral structure variations and random heterogeneities on AVO in sedimentary rocks are presented and analyzed Relection coeicients o sedimentary interaces Relection coeicients vary as a unction o oset rom the source (or relection angle) across an interace. his inormation is the core o AVO analysis. For either orward modeling or inversion, accurate calculation o relection coeicients is crucial. Here, we use the thin-slab method to calculate the relection coeicients at the shale/gas, shale/oil and shale/brine interaces. A rectangular grid o 14 is used in the calculation with its parameters given in able 1. he grid spacings used are 16 m in horizontal direction and 4 m in vertical direction. he interace is located at the depth o m. A taper unction is applied to the bottom o the model or eliminating the relection rom the artiicial truncation at bottom. Only P-P relection coeicients are displayed. For relection coeicient calculation, 49

50 we take samples (displacement amplitudes in space domain) in the middle o the model or both incident and relected waves, and deine the ratio o their average displacement amplitudes as the relection coeicient. Figure 1 shows the P-P relection coeicients or all sets o parameters given in able 1. he dotted curves represent the corresponding theoretical results.φ is the percentage porosity, and σ 1 and σ are the Poisson's ratios or the shale and sand, respectively. he top panel corresponds to shale/gas interace, the middle panel, shale/oil interace, and the bottom panel, shale/brine interace. We see that the relection coeicients calculated by the thin-slab method are in excellent agreement with theoretical values or small and medium incident angles (<3 o ) or all cases, and up to a wide angle o incidence (<4 o ) or the -percent-porosity gas, oil, and brine sands. Although the -percent-porosity sand has relatively large velocity perturbations, the thin-slab results show better matches to theoretical relection coeicients or wide angles o incidence than those or 3-percent-porosity and 5-percent-porosity sands at wide angles. It implies that the accuracy and wide-angle capacity o the thin-slab method are related not only to velocity and density perturbations but also to their combinations (the impedance). he computational eiciency, accuracy, and wide-angle capacity o the thin-slab method are closely related to the amount o perturbations. Generally, the accuracy and wide-angle capacity decrease as perturbations (velocity and/or density) increase. Its computational eiciency also decreases because iner orward steps must be utilized to guarantee the convergence. o see the perturbations o reservoir parameters shown in able 1, we calculate velocity and density luctuations relative to the shale (shown in able ). he perturbations o the physical parameters used in the models are typical o most sedimentary rocks. 5

51 6.. Relections rom a dipping sandstone reservoir Figure 13 shows a model o a dipping sandstone reservoir bearing gas, oil, and brine. he dipping angle o the reservoir is 1 o to horizontal plane. he reservoir is thick enough so that the relections rom the top and base are separated in seismograms. Simmons and Backus (1994) used these models to investigate AVO responses associated with angles o incidence, dierent type interaces, and porosities o interest. In this section we will use these models to show the accuracy o the thin-slab method or modeling relections including primary relections, conversions and diractions. Figure 14 displays the synthetic seismograms with incident plane waves or three dierent porosities (labeled in each panel). he plane P-wave is vertically downward incident to the model. he let column corresponds to the inite-dierence results and the right panel, to the thin-slab results. he last two arrivals are the converted shear waves produced at the top and base o the reservoir and propagated in the shale. Single-leg (one converted wave path in the layer) and double-leg (two converted wave paths in the layer) converted shear waves are weak at a small angle o incidence and overlap with the primary relections rom the base. Note that in the thin-slab results (the right column in Figure 14) the multiples within the sandstone are neglected. Comparing the two columns, we see that the thin-slab results are in good agreement with the inite-dierence results, implying that the multiples are not signiicant in sedimentary rocks Reservoir relections with scattering and attenuation rom a heterogeneous overburden We examine the eect o scattering and attenuation associated with heterogeneities in sedimentary rocks on AVO using the thin-slab modeling method. First we study the eects o 51

52 random scattering. he reservoir is a lat model generated by rotating the model in Figure 13. A -D random ield with exponential correlation unction is used to perturb the velocity and density parameters o the sedimentary rocks (Figure 15). he correlation lengths o the random ield are 1 m in horizontal direction and 4 m in depth. he rms values used are 1%, % and 3%. Note that both P- and S-wave velocities have the same distributions and rms perturbations; while the density has the same distribution but only one hal o the rms value o velocity perturbation. Only velocity rms values are indicated in the text and igures. For simplicity, we consider a plane P-wave vertically incident on the reservoir. Figure 15 shows the model and the snapshots at t =. s,.4 s and.6 s, respectively. he porosity o the sand is 5% and rms velocity perturbation is %. In Figure 15 we see that abundant coda waves are produced. he waveronts o both orward and relected waves are distorted. For the relected waves rom shale/oil and shale/brine interaces, serious distortions can be seen. Figure 16 shows the maximum magnitudes o the responses rom the top interace o the sand. he vertical axis is the logarithmic amplitude, and (a), (b) and (c) correspond to porosities o %, 3%, and 5%, respectively. Solid lines correspond to calculations in models without overburden heterogeneities, in which the luctuations in amplitudes are caused by the intererence o boundary diractions. he dotted, dashed and dotted-dashed lines correspond to overburden models having rms random velocity luctuations o 1%, % and 3%, respectively. he vertical interaces separating gas, oil and brine are indicated by arrows. Note that or an overlying shale with heterogeneities as small as rms=1%, the piece-wise uniorm relection amplitudes become luctuating due to the ocusing and deocusing o waves. As the velocity luctuation increases, stronger amplitude luctuations are generated. 5

53 he amplitude luctuations are closely related to the statistical properties o heterogeneities. he existence o a laterally varying overburden layer has serious eects on relecting waves rom reservoirs. It generates scattered waves and aects the relection characteristics o local interaces. For weak relection sands, the scattering eect rom heterogeneous overburden could be important and must be taken into account or AVO analysis. he top panel shows reservoir model bearing gas, oil and brine, respectively. he ormation is anelastic and heterogeneous. he lower three panels show AVO s or various kinds o interaces: (a) shale/gas, (b) shale/oil, and (c) shale/brine. For each kind o interace, three dierent constant Q s (Q = ininity, 15, 5) are given to shale. he sand has constant Qp = Qs = 1. he correlation lengths o the random ield or perturbing Q and elastic parameters are the same. he rms values are 4% or elastic parameters and 5% or Q. Next, we show combined eects o random scattering and intrinsic attenuation. In practice, the geologic models may contain arbitrary spatial variations in compressional- and shear-wave quality actors, as well as density and velocities (see the top panel in Figure 17). For each kind o interace, three dierent averaged Q s (i.e., Q=5, 15, ininity) are given to shale, and the sands have quality actors o Q P = Q S = 1. he correlation lengths o the random ield or perturbing Q and elastic parameters are the same. he rms values are 4% or elastic parameters and 5% or Q. the source and receiver array are located in shale and 1 m away rom the interace. he dotted lines in Figure 17 correspond to the homogeneous cases with constant Q s. We see that intrinsic attenuation mainly aect the absolute relected amplitudes and heterogeneities in both Q and elastic parameters aect local amplitude 53

54 luctuation with oset. In summary, the AVO responses o the target subsurace have been signiicantly deormed due to both the heterogeneities and intrinsic attenuation hin layer AVO response he amplitude response o a thin-bed has drawn increasing interest in hydrocarbon interpretation because large quantities o gas reserves are ound to be trapped with thin sands. he AVO response o a thin-bed is dierent rom that o a thick bed because o the eects o wave intererence, conversion and tunneling. For a thin-bed with thickness much less than the predominant wavelength, the grid methods such as inite-dierence and inite-element methods are too costly or modeling relections in AVO analysis. We investigate the ability o the thin-slab method or handling elastic thin-bed relections. he thin-bed used is 5 m thick, located at a depth o 15 m and is illed with an oil sand o porosity %. he predominant wavelength is 15 m, being 5 times greater than the thickness o the thin-bed. he surrounding medium is shale. Source and receivers are on the top o the model. For the calculation using the thin-slab method, the grid spacing in the horizontal direction is 1 m and in the vertical direction is 1 m or the background area and.5 m or the thin-bed. Figure 18 displays relection seismograms calculated by the relectivity method (a) and thin-slab method (b). NMO corrections are applied to the top o the sand and reduced time is used in showing seismograms. In Figure 18, A represents relections rom the top o the model, B gives relections rom both the top and base but no converted waves are involved, C includes all waves (exact solution) and D gives relections rom both the top and base including both single-leg and double-leg converted shear waves. Comparing A and B, we see 54

55 the big dierence between AVOs o a single impedance interace and two closely located impedance interaces. B is equivalent to primary-only relections. Comparing B with C and D, we see the important eect o locally converted shear waves that can alter amplitude variation with oset. Comparing C and D, only small dierences exist at large osets. In this case, the eect o multiples is negligible. For a homogeneous thin-slab under small-angle approximation, equation or one-return relection can be simpliied into (Wu, 1996; Xie and Wu, 1) P δ Zα U ( K ) = ik z u ( K ), Z α α α where Z α is the P-wave impedance. he above equation presents not only the amplitude o the thin-bed relection but also the change in wavelet. hereore, the thin-slab propagators represent the response o an elastic heterogeneous thin-layer. Since the choice o z or step interval in waveield extrapolation is lexible and can vary according to local heterogeneities, the thin-slab method can naturally handle arbitrarily thin layers AVO response in laterally varying media In this section we use a simple model containing both a truncated salt layer and a thin gas sand, which is located m below the salt layer (Figure 19a), to examine the AVO response using the thin-slab method. he parameters or the gas sand are given in able 1 and the corresponding porosity is %. he parameters or salt arev p = 4.48km / s, V s =.594km / s, ρ = 3.1g / cm. he thicknesses o the salt layers are m, 4 m and 8 m, respectively. he grid spacing is the same as that used in Figure 18 except or inside the salt body, where a.1m grid space is used. In Figure 19, the maximum negative amplitudes o the thin-bed responses are picked and 55

56 plotted versus oset. he solid curve represents the thin-bed (without salt) AVO. he AVO trend o a thin-bed is dierent rom that shown in Figure 1 (shale/gas, φ = ) where the relection coeicient increases with incident angles up to 45 o or an oset o 3 km. he dotted, dashed and dotted-dashed curves correspond to salt layer thicknesses o m, 4 m, and 8 m, respectively. We see that AVOs are altered due to the presence o the salt layer. he salt layer can cause diraction, deocusing, conversion and transmission loss. he diraction aects AVO by causing intererence between diracted waves rom the let side o the salt layer and relected waves rom the thin-bed at receiver locations. his intererence causes local variations in AVO measurements. he transmission loss aects all the relections passing through the salt body and causes a systematic decrease in AVO. Figure shows the combined eects o a truncated salt layer and random heterogeneities on AVO. Heterogeneities are introduced into the model shown in Figure 19a. he correlation lengths o the D random ield are 1 m in horizontal direction and 4 m in vertical direction. he rms luctuations used are 1%, %, and 3%, respectively. he our panels in Figure correspond to the cases o zero-salt, m, 4m and 8m-thick salt (all labeled in the igure). he case o zero-salt shows the eect rom heterogeneities alone. he ocusing and deocusing o the spatially correlated heterogeneities produce the local variation in relected amplitudes versus oset which becomes signiicant or sedimentary models with weak relections. Interpretation o AVO observations based on homogeneous elastic models will thereore bias rom actual properties o the target. he requency and amplitude variations o relections are closely related to the source spectrum and the statistical properties o velocity perturbations, although the overall AVO trends are still controlled by 56

57 the target properties and overburden structures. o improve AVO analysis in sedimentary rocks, it is necessary to have suicient acquisition apertures and take into account o the overburden structures including random heterogeneities and thin-bed eects. 7. CONCLUSIONS In this chapter renormalized MFSB (multiple-orescattering single-backscattering) equations and the dual-domain expressions or scalar, acoustic and elastic waves are treated in a uniied approach. he De Wol approximation neglects the reverberations (internal multiples) inside, but can model all the orward scattering phenomena, such as ocusing/deocusing, diraction, reraction, intererence, etc., and the primary relections. he De Wol approximation can be considered as the irst order term in a De Wol multiple scattering series. his single backscattering signal deined this way is the primary relections rom the real medium, and is dierent rom the Born approximation where the incident ield and the Green s unction are both deined in the reerence medium (a homogeneous medium). It is also dierent rom the irst term o the generalized Bremmer series, which uses a high-requency asymptotic solution (a WKBJ-like solution) as the Green s unction (in an equivalent reerence medium). wo versions o the one-return method (using MFSB approximation) are given: One is the wide-angle dual-domain thin-slab approximation; the other is the screen approximation. he latter involves a small-angle approximation or the wave-medium interaction. Q-actor rom intrinsic attenuations has been incorporated into the algorithm by the introduction o complex velocities. he relectivity method has also been incorporated to the thin-slab modeling to increase the eiciency. he theory and methods are applied to the ast calculation o synthetic seismograms. For 57

58 weak heterogeneities ( ± 15% perturbation), good agreement between the one-return method and inite dierence simulations veriies the validity o the one-return approach. However, the one-return approach is about -3 orders o magnitude aster than the elastic FD algorithm. he method can be applied to the ast modeling o AVA responses or a complex reservoir with heterogeneous overburdens. he method can be applied to most o the real cases where the perturbations o P- and S-wave velocities are around or smaller than 3%. he inluences o heterogeneous or random overburdens, thin-bedding, Q-variation, irregular salt layer, etc., can be studied using this modeling technique. Acknowledgements Helpul discussions with and comments rom M. Fehler, L.J. Huang, R. Hobbs, S. Jin, C. Mosher, and K. Wapenaar are greatly appreciated. his work was supported by the Department o Energy through various contracts and by the WOPI (Wavelet ransorm on Propagation and Imaging or seismic exploration) Research Consortium at the University o Caliornia, Santa Cruz. Facility support rom the W.M. Keck Foundation is also acknowledged. Contribution number 488 o CSIDE, IGPP, University o Caliornia, Santa Cruz. Reerences Aki, K. (1973). Scattering o P waves under the Montana Lasa. J. Geophys. Res. 78, Aki, K. and Richards, P. (198). Quantitative Seismology: heory and Methods, Vol. 1 and 58

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64 homson, C. J. (1999). he gap between seismic ray theory and ull waveield extrapolation. Geophys. J. Int. 137, homson, C.J. (5). Accuracy and eiciency considerations or wide-angle waveield extrapolators and scattering operators. Geophys. J. Int. 163, Van Stralen, M.J.N., de Hoop, M. and Blok, H. (1998). Generalized Bremmer series with rational approximation or the scattering o waves in inhomogeneous media. J. Acoust. Soc. Am. 14, Wales, S.C. (1986). A vector parabolic equation model or elastic propagation, in Ocean Seismo-Acoustics (ed. Akal and Berkson). Plenum, New York. Wales, S.C., and McCoy, J.J. (1983). A comparison o parabolic wave theories or linearly elastic solids. Wave Motion, 5, Wapenaar, C.P.A. (1996). One-way representation o seismic data. Geophys. J. Int. 17, Wapenaar, C.P.A. (1998). Reciprocity properties o one-way propagators. Geophysics 63, Wapenaar, K., van Wijngaarden, A. J., van Geloven, W., and van der Leij, E. (1999). Apparent AVA eects o ine layering. Geophysics 64, Widess, M.B. (1973). How thin is a thin-bed. Geophysics 38, Wild, A.J., Hobbs, R.W. and Frenje, L. (). Modeling complex media: an introduction to the phase-screen method. Phys. o Earth and Planet. Inter. 1, Wild, A.J., and Hudson, J. A. (1998). A geometrical approach to the elastic complex screen. J. Geophys. Res. 13,

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66 III, SPIE 571, Wu, R.S., Huang, L.J., and Xie, X.B. (1995). Backscattered wave calculation using the De Wol approximation and a phase-screen propagator. Expanded Abstracts, SEG 65th Annual Meeting, Wu, R. S., and Jin, S. (1997). Windowed GSP (generalized screen propagators) migration applied to SEG-EAEG salt model data. Expanded Abstracts, SEG 67th Annual Meeting, Wu, R.-S., and oksöz, M.N. (1987). Diraction tomography and multisource holography applied to seismic imaging. Geophysics 5, Wu, R. S., Jin, S., and Xie, X. B. (a). Energy partition and attenuation Lg waves by numerical simulations using screen propagators. Phys. Earth and Planet.Inter. 1, Wu, R. S., Jin, S., and Xie, X. B. (b). Seismic wave propagation and scattering in heterogeneous crustal waveguides using screen propagators: I SH waves. Bull. Seism. Soc. Am. 9, Wu, R.S., Luo, M., Chen, S. and Xie, X.B. (4). Acquisition aperture correction in angle-domain and true-amplitude imaging or wave equation migration. Expanded abstracts, SEG 74th Annual Meeting, Wu, X. Y., and Wu, R. S. (1999). Wide-angle thin-slab propagator with phase matching or elastic modeling. Expanded abstracts, SEG 69th Annual Meeting, Wu, R.S., and Xie, X.B. (1993). A complex-screen method or elastic wave one-way propagation in heterogeneous media. Expanded Abstracts, 3rd International Congress 66

67 o the Brazilian Geophysical Society. Wu, R.S. and Xie, X.B. (1994). Multi-screen backpropagator or ast 3D elastic prestack migration. Mathematical Methods in Geophysical Imaging II SPIE 31, Wu, X. Y., and Wu, R. S. (1). Lg-wave simulation in heterogeneous crusts with surace topography using screen propagators. Geophys. J. Int. 146, Wu, X.Y. and Wu, R.S., (3a). Fast modeling o D/3D elastic relections using thin-slab method. Expanded abstracts, SEG 73rd Annual Meeting, Wu, X.Y. and Wu, R.S., (3b). Synthesizing AVO responses in visco-elastic media using ast one-way elastic propagators. Expanded abstracts, SEG 73rd Annual Meeting, 8-1. Wu, X. Y. and Wu, R.S. (5). AVO modeling using elastic thin-slab method. Geophysics, in press. Xie, X.B. and Wu, R.S. (1995). A complex-screen method or modeling elastic wave relections, Expanded Abstracts, SEG 65th Annual Meeting, Xie, X.B., and Wu, R.S. (1998). Improve the wide angle accuracy o screen method under large contrast. Expanded abstracts, SEG 68th Annual Meeting, Xie, X. B., Mosher, C. C., and Wu, R. S. (). he application o wide angle screen propagator to D and 3D depth migrations, Expanded Abstracts, SEG 7th Annual Meeting, Xie, X. B., and Wu, R. S. (1). Modeling elastic wave orward propagation and relection using the complex-screen method. J. Acoust. Soc. Am., 19, Xie, X. B., and Wu, R. S., (5). Multicomponent prestack depth migration using elastic 67

68 screen method. Geophysics 7, S3-S37. Yoon, K.H. and McMechan, G.A. (1996). 3D eighth-order elastic inite-dierence modeling o reraction and strong-motion data rom the Coyote Lake region, Caliornia. Bull. Seis. Soc. Am., 86,

69 Figure captions Figure 1: Sketch showing the meaning o the De Wol (one-return) approximation. Figure : Schematic illustration o the thin-slab method or implementing the one-return approximation: (a) Original medium is sliced into thin-slabs, (b) iterative procedure o transmitted and relected wave calculations by the one-return approximation. Figure 3: Geometry or the derivation o the thin-slab method. Figure 4: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he upper panel corresponds to P wave and the lower panel to S wave incidences respectively. he bottom layer o the model has 1% velocity perturbations or both P and S waves with respect to the top layer. Figure 5: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he perturbation o the bottom layer is 1% with respect to the top layer or both P and S wave velocities. Figure 6: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he perturbation o the bottom layer is % with respect to the top layer or both P and S wave velocities. 69

70 Figure 7: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he perturbation o the bottom layer is % with respect to the top layer or both P and S wave velocities. Figure 8: A -D heterogeneous model (French, 1974) with irregular interace used to test the validity and accuracy o the thin-slab method. he background medium has α = 3.6km / s, β =.8km / s and 3 ρ =.g / cm. he layer in black color has a perturbation o % or both P and S wave velocities. Figure 9: Comparison o synthetic seismograms calculated by inite dierence (solid) and by thin-slab methods (dashed). Curves in (a) and (b) are horizontal and vertical components o displacement, respectively. Figure 1: he 3D French model or the numerical tests. he at solid lines delineated the horizontal surace at which the thin-slab method is connected to the relectivity method. he background medium has the same parameters as those in Figure 8. he structure in grey color has a perturbation o -1% or both P and S velocities. Figure 11: Comparison o synthetic seismograms calculated by inite dierence (solid lines) and by thin-slab methods (dashed lines). From top to the bottom are X-, Y- and Z-components o displacement, respectively. 7

71 Figure 1: P-P relection coeicients at dierent type o interaces: shale/gas (top), shale/oil (middle), and shale/brine (bottom). All ormation parameters are listed in able 1. he solid curves are calculated by the thin-slab method and the dotted lines, by the Zoeppritz equations. Figure 13: Model o a dipping sandstone reservoir illed with gas, oil, and brine. he reservoir is thick enough so that the relections rom the top and base can be separated. Figure 14: Comparisons o plane wave seismograms calculated by inite-dierence (let column) and thin-slab (right column) methods or a dipping reservoir model shown in Figure 13. Figure 15: (a) Sandstone model with heterogeneities, (b) to (d) are Snapshots at t =. s,.4 s, and.6 s, respectively. A 3 Hz plane P-wave source is vertically incident rom the top o the model. Figure 16: Eect o scattering by heterogeneities on relected amplitudes o reservoirs shown in Figure 15. Figure 17: he top panel shows reservoir model bearing gas, oil and brine, respectively. he ormation is anelastic and heterogeneous. he lower three panels show AVO s or various kinds o interaces: (a) shale/gas, (b) shale/oil, and (c) shale/brine. For each kind o interace, 71

72 three dierent constant Q S (Q = ininity, 15, 5) are given to shale. he sand has constant Q P = Q S = 1. he correlation lengths o the random ield or perturbing Q and elastic parameters are the same. he rms values are 4% or elastic parameters and 5% or Q. Figure 18: Relection seismograms or an oil sand model. he oil sand is 5 m thick and at depth o 15 m. (a) By relectivity method, (b) by thin-slab method. In the igure, A represents relections rom the top o the model (thick layer), B, relections rom both top and base but no converted waves included, C, all waves included (exact solution), and D, relections rom both top and base including single-leg and double-leg converted shear waves. Figure 19: (a) Model containing a truncated salt layer with thickness d and a thin gas sand layer below the salt, (b) Angle-dependent relections (AVO) o a thin gas layer with relections rom the salt layer o dierent thickness. Figure : AVO s o the thin gas layer with the combined eects o lateral structure variation and random heterogeneities. 7

73 able 1. Reservoir model. φ = % φ = 3% φ = 5% α β ρ σ α β ρ σ α β ρ σ shale gas oil brine able. Perturbations o reservoir parameter. φ = % φ = 3% φ = 5% δα α % δβ β % δρ ρ % σ δα α % δβ β % δρ ρ % σ δα α % δβ β % δρ ρ % σ gas oil brine

74 Figure 1: Sketch showing the meaning o the De Wol (one-return) approximation. 74

75 Figure : Schematic illustration o the thin-slab method or implementing the one-return approximation: (a) Original medium is sliced into thin-slabs, (b) iterative procedure o transmitted and relected wave calculations by the one-return approximation. 75

76 Figure 3: Geometry or the derivation o the thin-slab method. 76

77 Relection coe. (P-P) (P-P) heory hin-slab o Incident angle ( ) Relection coe. (P-S) (P-S) heory hin-slab Incident angle ( o). (S-P). (S-S) Relection coe. (S-P) heory hin-slab o Incident angle ( ) Relection coe. (S-S) heory hin-slab Incident angle ( o) Figure 4: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he upper panel corresponds to P wave and the lower panel to S wave incidences respectively. he bottom layer o the model has 1% velocity perturbations or both P and S waves with respect to the top layer. 77

78 Relection coe. (P-P) (P-P) heory hin-slab o Incident angle ( ) Relection coe. (P-S) (P-S) heory hin-slab o Incident angle ( ). (S-P). (S-S) Relection coe. (S-P) heory hin-slab o Incident angle ( ) Relection coe. (S-S) heory hin-slab Incident angle ( o) Figure 5: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he perturbation o the bottom layer is 1% with respect to the top layer or both P and S wave velocities. 78

79 Relection coe. (P-P) (P-P).8.6 heory.4 hin-slab o ( a ) Incident angle ( ) Relection coe. (S-P) (S-P)..1 heory. -.1 hin-slab o ( b ) Incident angle ( ) Relection coe. (S-S) Relection coe. (P-S) (P-S)..1 heory. -.1 hin-slab Incident angle ( o) (S-S)..1 heory. -.1 hin-slab Incident angle ( o) Figure 6: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he perturbation o the bottom layer is % with respect to the top layer or both P and S wave velocities. 79

80 Relection coe. (P-P) (P-P). heory. -. hin-slab o ( a ) Incident angle ( ) Relection coe. (S-P) (S-P)..1 heory. hin-slab ( b ) o Incident angle ( ) Relection coe. (P-S) Relection coe. (S-S) (P-S). heory.1. hin-slab o Incident angle ( ) (S-S). heory hin-slab Incident angle ( o) Figure 7: Comparisons o relection coeicients calculated by the thin-slab method and the exact solutions. he perturbation o the bottom layer is % with respect to the top layer or both P and S wave velocities. 8

81 Figure 8: A -D heterogeneous model (French, 1974) with irregular interace used to test the validity and accuracy o the thin-slab method. he background medium has α = 3.6km / s, β =.8km / s and 3 ρ =.g / cm. he layer in black color has a perturbation o % or both P and S wave velocities. 81

82 Figure 9: Comparison o synthetic seismograms calculated by inite dierence (solid) and by thin-slab methods (dashed). Curves in (a) and (b) are horizontal and vertical components o displacement, respectively. 8

83 Figure 1: he 3D French model or the numerical tests. he at solid lines delineated the horizontal surace at which the thin-slab method is connected to the relectivity method. he background medium has the same parameters as those in Figure 8. he structure in grey color has a perturbation o -1% or both P and S velocities. 83

84 Figure 11: Comparison o synthetic seismograms calculated by inite dierence (solid lines) and by thin-slab methods (dashed lines). From top to the bottom are X-, Y- and Z-components o displacement, respectively. 84

85 Figure 1: P-P relection coeicients at dierent type o interaces: shale/gas (top), shale/oil (middle), and shale/brine (bottom). All ormation parameters are listed in able 1. he solid curves are calculated by the thin-slab method and the dotted lines, by the Zoeppritz equations. 85

86 Figure 13: Model o a dipping sandstone reservoir illed with gas, oil, and brine. he reservoir is thick enough so that the relections rom the top and base can be separated. 86

87 Figure 14: Comparisons o plane wave seismograms calculated by inite-dierence (let column) and thin-slab (right column) methods or a dipping reservoir model shown in Figure

88 Figure 15: (a) Sandstone model with heterogeneities, (b) to (d) are Snapshots at t =. s,.4 s, and.6 s, respectively. A 3 Hz plane P-wave source is vertically incident rom the top o the model. 88

89 Figure 16: Eect o scattering by heterogeneities on relected amplitudes o reservoirs shown in Figure

90 Figure 17: he top panel shows reservoir model bearing gas, oil and brine, respectively. he ormation is anelastic and heterogeneous. he lower three panels show AVO s or various kinds o interaces: (a) shale/gas, (b) shale/oil, and (c) shale/brine. For each kind o interace, three dierent constant Q S (Q = ininity, 15, 5) are given to shale. he sand has constant Q P = Q S = 1. he correlation lengths o the random ield or perturbing Q and elastic parameters are the same. he rms values are 4% or elastic parameters and 5% or Q. 9

91 Figure 18: Relection seismograms or an oil sand model. he oil sand is 5 m thick and at depth o 15 m. (a) By relectivity method, (b) by thin-slab method. In the igure, A represents relections rom the top o the model (thick layer), B, relections rom both top and base but no converted waves included, C, all waves included (exact solution), and D, relections rom both top and base including single-leg and double-leg converted shear waves. 91

92 Figure 19: (a) Model containing a truncated salt layer with thickness d and a thin gas sand layer below the salt, (b) Angle-dependent relections (AVO) o a thin gas layer with relections rom the salt layer o dierent thickness. 9