Summary. Introduction

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1 Imae amplitude compensations: propaator amplitude recovery and acquisition aperture correction Jun Cao and Ru-Shan Wu Department of Earth and Planetary Sciences/IGPP, University of California, Santa Cruz Summary Many factors may influence the amplitude of mirated imae. The acquisition aperture correction and propaator eometrical spreadin correction (propaator amplitude recovery) have shown to be important for improvin the mirated imae. Usin local-anle domain one-way wave equation based miration, we investiate the influence of these two factors on the imae amplitude to better understand their relative importance in true-reflection/amplitude imain. We utilize the localized WKBJ correction in eneral heteroeneous media to recover the true eometrical spreadin of one-way wave propaator. We then apply this localized WKBJ solution to miration and compare its effect with that of the acquisition aperture correction on the imae amplitude. Numerical examples indicate that the effect of the propaator WKBJ compensation on the imae amplitude is less sinificant than that of the acquisition aperture correction for the miration with limited acquisition aperture in eneral media. Introduction Seismic miration operators are not the inverse of the forward modelin operator due to many factors, which we will discuss in the followin. As a result, imaes produced by miration have artifacts and biased amplitudes. Truereflection/amplitude imain tries to ive not only correct location but also correct imae amplitude (or AVA) of the reflectors or direct estimation of the media property. When waves propaate throuh the anelastic heteroeneous Earth, propaation effects (e.., eometric spreadin, transmission loss, intrinsic attenuation) chane the amplitude of the wavefield. Due to limited data acquisition aperture in media with complex overburden structures, seismic miration/imain often provides stronly non-uniform imae amplitude of subsurface structures. To obtain a truereflection/amplitude imae, we should take all of these effects into consideration. True-reflection imain has been and remains a challene in seismic processin. We focus on wave-equation based methods here. Geometrical spreadin is automatically handled if the wavefield extrapolation uses full-wave equations based methods (e.., Den and McMechan, 27); however the full-wave equations based propaator is computationally much more expensive than the one-way propaator. Conventional one-way propaators do not treat eometric spreadin correctly. The asymptotic true-amplitude one-way propaator can provide accurate amplitude in the sense of hih-frequency approximation (e.., Zhan, 1993). With this propaator, better imae amplitude is obtained in some smoothly varyin models usin the sinle-shot division-type imain condition (Zhan et al., 23; 25); however only eometric spreadin is handled in that method. The limited data acquisition aperture in media with complex overburden structures often results in stronly non-uniform dip-dependant illumination and distorted imae amplitude of subsurface structures. Note that U/D shot-profile imain conditions based on deconvolvin the source side downoin wave (D) from the receiver side upoin wave (U) (e.., Zhan et al., 25; Den and McMechan, 27) do not take into account the influence of finite recordin eometry. Wu et al. (24) propose an amplitude correction method in the local anle domain for acquisition aperture effects which include the acquisition confiuration and propaation path effects throuh complex overburdens. It can sinificantly improve the imae amplitude in prestack depth miration. The methods discussed above use the back-propaation interal based imain condition. There is another type of potential true-reflection imain method based on minimizin the misfit function usin the adjoint operator in the Hessian matrix (Lailly 1983; Tarantola 1984). We will discuss it in a separate paper. The acquisition aperture correction and propaator eometrical spreadin correction have shown to be important for improvin the mirated imae. Here we compare their effects on the imae amplitude to understand their roles in the true-reflection imain. First, we summarize the theory of localized WKBJ solution for one-way propaator eometrical spreadin correction in heteroeneous media. Then we ive the acquisition-aperture correction formulas for different imain conditions. Finally, we apply the local WKBJ corrected propaator to the miration and compare the influence of the propaator amplitude correction with that of the acquisition aperture correction on imae amplitude. WKBJ solution in the local wavenumber domain for oneway wave propaators Most of the amplitude compensation schemes for one-way propaators are formulated and implemented in the space domain (e.., Zhan et al., 25). The localized WKBJ correction for eneral heteroeneous media (Wu and Cao,

2 25; Luo et al., 25) is in the local anle/wavenumber domain. In eneral heteroeneous media, it is hard to define lobal wavenumbers, so beamlet method (e.., Wu et al., 2), which contains localized wavenumber and location information, could be used to implement the localized WKBJ correction (Luo et al., 25). The WKBJ solution in smoothly varyin v(z) media (e.., Stolt and Benson, 1986) can be derived from enery flux conservation (see Wu and Cao, 25), P2 cos θ1 ρ2v2 ρ2 kz( v1) = =, (1) P1 cos θ2 ρ1v1 ρ1 kz( v2) where P and ρ are pressure and media density respectively; θ is the propaation anle with respect to the i vertical direction and k z is the vertical wavenumber. We can eneralize the WKBJ solution for smoothly varyin v(z) media to a transparent (or enery-conservative) propaator for eneral v(z) media. For v(z) media with discontinuities, the WKBJ solution 1 corresponds to a transparent boundary condition at each interface and the propaator becomes a transparent propaator, which nelects all of the scattered/reflected enery loss when the waves cross the boundary. The enery flow is thus continuous and conserved in both the smoothly varyin media and the media with sharp boundaries. Alon this line, the concept of transparent propaator can be extended to eneral heteroeneous media in the local anle/wavenumber domain (Wu & Cao, 25). It can be called the localized WKBJ correction. It can be implemented usin a beamlet propaator (Luo et al., 25). Accordin to the above lobal WKBJ correction 1, the localized WKBJ correction in eneral heteroeneous media can be symbolically written utilizin the beamlet (b mn ) propaator (e.. Wu et al., 2) as u( xn, ξm, z+δ z, ω) ρ( xn, z+δz) kz( xn, z) =, (2) ux ( n, ξm, z, ω) ρ( xn, z) kz( xn, z+δz) with kz( xn, z) = ω v ( xn, z) ξ (3) m as the window-position dependent local vertical wavenumber at depth z, where ux ( n, ξm, z, ω ) is the beamlet domain wavefield, vx ( n, z ) is the reference velocity for window at x, and n ξ m is the wavenumber. The calculated impulse responses in eneral heteroeneous media demonstrate the effect of the localized WKBJ correction on the wavefield amplitude (Luo et al., 25). Acquisition aperture correction In this part, we ive the acquisition-aperture correction formulas for the crosscorrelation and division type imain conditions. For detailed derivation, please refer to Wu et al. (24). (1) For crosscorrelation type imain condition The imain condition in the local anle domain (e.., Wu and Chen, 22; Chen et al., 26) can be written as, L( x, θs, θ) = 2 GI( x, θs; xs) xs, (4) GI( x, θ; x) dx D( x; xs) A( x; xs) z where GI ( x, θs; x s) is the incident wavefield in the local anle domain, the interal is the back-propaated wavefield in the local anle domain from the recorded data D (x ; x s ), A(x ; x s ) is the receiver aperture for a iven source at x, s stands for complex conjuate, and θ s and θ are the source and receivin anles respectively. Then the imae obtained at each imain point is no loner a scalar but a matrix, called the local imae matrix L( x, θ, θ ). The local imae matrix is a distorted estimate of the local scatterin matrix due to the acquisition aperture limitation and the propaation path effects. Local scatterin matrix is the intrinsic property of the scatterin medium and contains information of the local structure and elastic properties (Wu et al., 24). It is independent of the acquisition system and free from propaation effects. The task of true reflection/amplitude imain is to restore the true local scatterin matrix from the distorted local imae matrix by applyin imae amplitude corrections. Wu et al. (24) proposed an amplitude correction method in the local anle domain for the acquisition aperture effect. The relevant amplitude correction factor matrix, F a, for above imain condition 4 is, 1/2 2 2 F ( x, θ, θ ) = G ( x, θ ; x ) dx G ( x, θ ; x ). x x s { A( ; ) } a s I s s I s xs (5) The final acquisition aperture corrected imae can be obtained by applyin the amplitude correction factor to the miration imae in the local anle domain. (2) For division type imain condition The U/D shot-profile imain conditions based on deconvolvin the source side downoin wave (D) from the receiver side upoin wave (U) are used to seek a trueamplitude imae (e.., Zhan et al., 25; Den and McMechan, 27). Althouh these imain condition do not take into account the influence of finite recordin eometry, they miht ive imae amplitude more close to the reflection strenth than the crosscorrelation type imain conditions. We will use this imain condition for the followin sinle shot miration in a simple model. However, we should keep in mind that the division imain condition is less stable than the crosscorrelation type imain condition since the correction is done before the stack over the shots; therefore for eneral case we still use the crosscorrelation type imain condition. For the sinle shot problem, the division imain condition to obtain the scatterin strenth in local anle domain can be written as,

3 G L( x, θs, θ) = 2 G A( x ; x s) dx I( x, θs; xs) 2 I( x, θs; xs) GI( x, θ; x) us x xs ( ; ). (6) z By similar derivation to the crosscorrelation type imain condition, we can obtain the correspondin acquisition aperture correction factor F a for imain condition 6, 2 Fa( x, θs, θ) = dx (, ; ) A( ; s) GI θ x x. (7) x x Comparison of the effects of propaator amplitude correction and acquisition aperture correction In this part, we investiate the influence of the propaator eometrical spreadin correction and the acquisition aperture correction on the imae amplitude usin numerical examples in a smoothly varyin v(z) and a eneral heteroeneous medium to understand their roles in the true-reflection imain. (1) v(z) model First, a simplified example is desined to illustrate the eneral idea. A point scatterer is put in a smoothly varyin v(z) medium. A split-spread survey covers the surface and the source is riht above the scatterer. The velocity model used is v(z)= z (km/s). Since there is only one incident anle θ for the example here, we investiate s = the imae only in the receivin/scatterin anle θ domain at the scatterin point x, L( x,, θ ). Fiure 1 shows the imae amplitude with different corrections for data with 6 km aperture. For the miration with the WKBJ-corrected one-way propaator, the imae amplitude curve within ±2 is flat, which arees with the theoretical prediction. Without the WKBJ correction, the imae amplitude is smaller and decreases faster with the increase of the scatterin anle. The imae amplitude after the aperture correction sinificantly improves for lare scatterin anles compared with that before the aperture correction for the conventional propaator. The aperture correction has a stroner compensation effect on the imae amplitude than the propaator WKBJ correction. The imae with both the WKBJ and aperture corrections shows no noticeable difference from that with only the aperture correction. (2) 2D SEG/EAGE salt model Here we utilize the local cosine basis beamlet propaator with the localized WKBJ correction in miration to compare the influence of the above two corrections on the imae amplitude for the 2D SEG/EAGE salt model (Aminzadeh et al., 1994; Aminzadeh et al., 1995) (Fiure 2). 1/2 Imae amplitude Scatterin anle ( o ) Fiure 1: Comparison of the influence from the WKBJ correction and acquisition aperture correction on the imae amplitude for the 6km split-spread acquisition. Dash-dot line: oriinal imae without any correction; dashed line: imae with the WKBJ correction only; dotted line: imae after the aperture correction only; solid line: imae with both the WKBJ and aperture corrections Fiure 2: 2D SEG/EAGE salt model with the rid size dx=dz =4 feet Fiure 3: Poststack miration imaes before ( and after ( the localized WKBJ correction. Here, dx=dz =4 feet. Post-stack miration is a wave back-propaation process; therefore it can directly show the effect of the WKBJ correction on the propaator amplitude. The post-stack miration imae amplitude after the localized WKBJ correction is stroner throuhout the model (Fiure 3), especially for the steep faults in the sediment, the salt boundary, and the subsalt structures. The reason is that the WKBJ solution represents a transparent propaation, which nelects all the reflection/scatterin loss durin extrapolation. The side effect is that the WKBJ correction miht increase the amplitude of the artifacts, especially within and below the salt dome here. Similar to the post-stack case, the pre-stack miration imae amplitude after the localized WKBJ correction with the crosscorrelation imain condition is stroner throuhout the

4 model and the imae has stroner artifacts too (Fiure 4). To make the imae more closely represent the reflection strenth, we normalize the imae with the source side illumination. This correction assumes the receiver aperture is infinite. The normalized imaes with/without the localized WKBJ correction ive very similar amplitude (Fiure 5). The small-offset acquisition (maximum offset = 14 ft) and relatively steep dippin structures in the SEG/EAGE salt model may be the reasons for the relatively weak effect of the WKBJ correction on the imae amplitude. The smalloffset acquisition can only acquire small anle reflections from the steeply dippin reflectors. This makes the paths of the incident and reflected waves quite similar. The top layer of the model is homoeneous water. Therefore the effects of the WKBJ correction for the incident and reflected waves are similar, which makes the imae amplitude improvement after the propaator correction less sinificant. The imae after the acquisition aperture correction (Fiure 6 improves sinificantly throuhout the model compared with the imae before the correction (Fiure 4. The imaes of the steep faults in the sediment are sharper and more continuous. For subsalt structures, the imaes alon the steep structures and the baseline are more uniformly distributed after the aperture correction. The artifacts in the subsalt reion caused by salt body related multiples are also reduced relatively. The imain with both the localized WKBJ and acquisition aperture corrections (Fiure 6 ives a result of similar structural imae to that with only the aperture correction (Fiure 6, but stroner artifacts in the subsalt area compared to the latter. These results show that the acquisition aperture correction has a stroner effect on the imae amplitude for the 2D SEG/EAGE salt model. This arees with the result in the above v(z) medium. As for the efficiency of the acquisition aperture correction, the oriinal implementation (Wu et al., 24) is rather timeconsumin since the wavefield is decomposed into the local anle domain usin local slant stack (e.., Xie and Wu, 22) which is computationally demandin. We (Cao and Wu, 28) proposed a faster implementation usin beamlet decomposition. The new method produces results similar to those by the oriinal local slant stack based method. However, it can be more than twice as fast as the local slant stack based method and only cost about twice the computation time of traditional one-way wave-equation based miration. Conclusions We investiate the influences of the acquisition aperture correction and one-way propaator eometrical spreadin correction on the imae amplitude utilizin the local anle domain one-way wave equation based miration. Throuh numerical examples, we come to the followin conclusions. First, the WKBJ correction for the propaator amplitude shows some improvements on imae amplitudes, but not sinificant, and the WKBJ correction miht brin a side effect, amplifyin the artifact. Second, the effect of the acquisition aperture correction on the imae amplitude is more sinificant than that of the propaator WKBJ compensation for the miration with limited acquisition aperture in eneral media. Acknowledements We would like to thank Xiao-Bi Xie for helpful discussions, and Minqiu Luo for helpful discussions and providin the local cosine basis propaation code with the local WKBJ correction Fiure 4: Prestack imaes before ( and after ( the localized WKBJ correction. Here, dx=dz=8 feet Fiure 5: Prestack imaes normalized with the source side illumination before ( and after ( the localized WKBJ correction. Here, dx=dz =8 feet Fiure 6: Prestack imaes with the acquisition aperture correction before ( and after ( the localized WKBJ correction. Here, dx=dz =8 feet.

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