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1 elocity analysis for plane-wave source migration using the finite-frequency sensitivity kernel Hui Yang, ormerly niversity of California at anta Cruz; presently LandOcean Energy ervices Co., Ltd., eijing, 184, China. Xiao-i Xie, and Yaofeng He, GPP, niversity of California at anta Cruz, anta Cruz, CA 9564, A. ummary We derive the finite-frequency sensitivity kernel which relating the errors in migration velocity model to residual moveout in common image gathers obtained from planewave source prestack depth migration. The broadband sensitivity kernel is validated by the directly measured sensitivity map. ased on this kernel, we propose a new wave-equation migration velocity analysis approach particularly for plane-wave source migration. This method avoids the expensive local angle or offset decomposition. Numerical method based on the one-way propagator is designed to calculate the sensitivity kernel. sing a simple synthetic data set, we demonstrate the potential application of this new method in migration velocity analysis. ntroduction Wave-equation based short-record prestack depth migration (PM) is widely used in reflection seismology for imaging the complex subsurface structures. However, shotrecord PM is usually less efficient. To overcome this disadvantage, the shot record can be converted into synthetic plane-wave source record for PM (Zhang, et al., 5; Liu, et al., 6; toffa, et al., 6). The related migration velocity analysis (MA) has been investigated by Ji (1997) based on the high frequency asymptotic method. Jiao et al. () conducted residual velocity analysis for plane-wave migration using a local 1- model. n general, the MA is still dominated by the ray-based method which is inconsistent with the wave-equation PM. The sensitivity of finite-frequency signal to velocity variation has been investigated and applied to seismic tomography (e.g., Luo and chuster, 1991; Woodward, 199; asco et al., 1995, petzler and nieder, 4; ava and iondi, 4; e Hoop, et al., 6; Jocker, et al., 6; liedner and evc, 8). ava and iondi (4) tested the wave-equation MA, and ava and lad (8) detailed its numerical implementation for zero-offset migration, survey-sinking migration and shot-profile migration. Xie and Yang (7, 8a, b) derived the sensitivity kernel which relating the observed residual moveout (RMO) in shot-index common image gather (CG) to the velocity model errors and proposed a MA for shotgather migration. n this paper, we derive the sensitivity kernel relating the RMO in surface plane-wave source CG to errors in velocity models. We then validate this kernel and propose a novel velocity updating approach particularly for PM with synthetic surface plane-wave sources. ensitivity Kernel for Plane-Wave ource RMO n a plane-wave source migration, the down-going wave u can be synthesized from individual shot ( rp ;, ) responses u ( ;, ) through (e.g., Zhang, et al., 5) rr i u ( ;, u( ;, e pr rp = rr dr, (1) r is the space location, r is the shot location, is the angular frequency, p= eˆ P sin ( αs) v is the wave s parameter vector of the synthetic plane-wave source, eˆ P is a unit vector indicating the bearing direction of the plane wave source, α s is the plane-wave take-off angle from vertical direction, and v s is the average near-surface velocity. imilarly, the up-going wave u from a ( rp ;, reflector can be obtained from up-going waves of individual shots u ) using the same phase delay ( rr ;, i u ( ;, ) u ( ;, ) e pr rp = rr dr. () Note, although labeled with the parameter p, the downgoing wave u is a plane wave only at the surface. oth u and are not plane waves in a general heterogeneous u model. Assuming we conduct the PM in an initial velocity model v () r which is sufficiently close to the true velocity model v, the MA is trying to determine the velocity T () r difference δ v= v from the inconsistency of the depth T v image. The partial image obtained in the initial model v for plane source p can be expressed as (e.g., Zhang et al. 5) ( r, p, = u( r; p, u ( r; p,, (3) u and are extrapolated down- and up-going u wavefields in the initial model, the asterisk denotes the complex conjugate, r is the location of the image point. The information regarding the velocity error δv (which 367

2 elocity analysis for plane-wave source migration can also be seen as the unknown velocity perturbation related to v ) is carried by down- and up-going waves entering the image as phase delay. sing Rytov approximation (e.g., laney, et al., 1984; Jocker, et al, 6; Xie and Yang, 8a), the phase delay can be expressed as ( r; p, ( r; p, δϕ ( r, p, = m +, (4) u( r; p, u ( r; p, and are scattered waves due to the unknown velocity perturbation δ v. ased on the orn approximation, these scattered waves can be expressed as ( ) ( ) r; p, = k m( r ) G( r ; p, G( r ; r, d (5) ( ) ( ) ( ) ( ) r; p, = k m r G r ; p, G( r ; r, d (6) m( r) = δv( r) v ( r) is the relative velocity k = v r is the background perturbation, ( r, ) ( ) ( ) G ( rr ;, G ( rp ;, G ( rp ;, wavenumber, is the source spectrum, is the Green s function, and are plane-wave Green s functions defined by u rp ;, = G rp ;,, (7) ( ) ( ) ( ) ( ) ( ) ( ) u rp ;, = G rp ;,. (8) All Green s functions are calculated in the initial model. The integral covers the entire space with velocity perturbations. ubstituting equations 5-8 into 4, we have, δϕ ( r ) ( ) ( ), p, = m r K r, r, p, d (9) + m( r ) K (,,, ) r r p d, G( rp ;, G( rr ;, K ( rr,, p, ) = m k,(1) G( r; p, ) and G ( rp ;, G( rr ;, K ( rr,, p, ) = m k,(11) G ( r; p, are single frequency sensitivity kernels or réchet kernels for the source and receiver sides. Equations 9-11 setup the relationship between the velocity perturbation and phase delay in the migrated image. ollowing Xie and Yang (8a), we can convert the phase delayδϕ into broadband traveltime delay δt, δ t( r, p) = m( r ) K( r, r, p) d (1) + m( r ) K ( r, r, p) d, ( ) W K ( rr,, p) = K ( rr,, p, d, (13) W ( K ( rr,, p) = K ( rr,, p, d, (14) are broadband traveltime delay sensitivity kernels for the source and receiver side, and W ( is a weighting function (Xie and Yang, 8a). igure. 1: alidation of sensitivity kernel for plane-wave source PM, (a) is the theoretical sensitivity kernel and (b) is the sensitivity map directly measured from migration image. The trial perturbation, shown as a small solid circle in (b), has a diameter of m and a velocity perturbation of 5%. The traveltime delay can be further converted into the RMO in depth domain via (e.g., tork, 199; Xie and Yang, 8a) R( r, p) = A( r, p) δt( r, p), (15) A( r, p ) = v ( r ) cos θ( r, p ) cosα( r ), (16) θ ( r, p) is the reflection angle relating to the reflector normal direction and α ( r ) is the target dip angle at image point r. ubstituting equations 1-14 into equation 15, we finally have the relationship between the velocity perturbation δ v and the RMO in depth image. The correspondent sensitivity kernel is for CGs indexed by the plane-wave sourcep. alidation of Theoretical ensitivity Kernel To validate the theoretical formulation, we compare the sensitivity kernel calculated from equations 1-16 with the directly measured sensitivity map. hown in ig. 1a is the broadband theoretical kernel calculated in a model which has a constant background velocity of 3.5 km/s and a horizontal reflector located at depth.5 km. The synthetic plane-wave source has an incident angle of 3 and a

3 elocity analysis for plane-wave source migration Hz Ricker wavelet. The image point is located at distance 5. km. A one-way generalized screen propagator based on the multiple-forward scattering and single back-scattering approximation (Xie and Wu, 1; Wu et al. 6) is used to calculate the broadband sensitivity kernel. ig. 1b shows the sensitivity map directly measured from plane-wave PM imaging. The method has been detailed in Xie and Yang (8a). The general consistency between these sensitivity kernels verifies the theory and algorithm. δ R ( r, p1, p) m( ) K (,, 1, ) d, (18) = r r r p p, K ( r, r, p1, p) = A( r, p) K ( r, r, p) K ( r, r, p) A( r, p1) K ( r, r, p1) K ( r, r, p1), (19) is the broadband differential sensitivity kernel which combines kernels from a pair of plane-wave parameters p and 1 p within the same CG. Equations 18 and 19 form the inversion system for the perturbation m() r which will be used to update the initial velocity model. llustrated in ig. are differential sensitivity kernels in a model, a is the heterogeneous velocity model, b is the theoretically calculated differential sensitivity kernel using equation 19 and a pair of plane waves of 45 and 3, and c is the directly measured sensitivity map. n general, igs. b and c have the similar pattern. igure : Comparison of differential sensitivity kernels in a heterogeneous background model, with (a) the velocity model, (b) the theoretically calculated differential kernel, and (c) the measured sensitivity map. The take-off angles of the two planewave sources are 45 and 3, respectively. igure 4: Result from plane-wave source migration in the initial model, with (a) stacked image, and (b) (e) CGs at selected image points indicated by crosses in (a). igure 3: elocity model used to test the MA. nversion ystem n practice, the absolute RMO is usually unavailable. nstead, MA often adopts the degree of alignment of different experiments within the same CG to judge the migration velocity model. The best alignment of events can be achieved by minimizing the relative RMO δ R( r, p1, p) = R( r, p) R( r, p 1). (17) ubstituting equations 1-16 into equation 17, δ R can be expressed as: igure 5: nverted velocity perturbations. Numerical Example llustrated in ig. 3 is a simple model adopted to demonstrate the potential applications of the plane-source sensitivity kernel in MA. The velocity model is composed 367

4 elocity analysis for plane-wave source migration of a 3.5 km/s constant background velocity and four patches of velocity perturbations in the middle of the model. The velocity perturbation in each patch is Gaussian distributed with its perturbation at the center is 1%. Their different sizes are used to test the inversion resolution. A horizontal reflector is located at depth.5 km. hot gather records with full acquisition aperture are synthesized using a fourth-order scalar wave finite-difference method. sing equations 1- and taking α, we compose the shot s = 45 gathers into 31 surface plane-waves, which are equally distributed in p values. sing a one-way wave equation based method (Wu, et al., 8), we first conduct the plane-wave source PM in the background model. hown in ig. 4a is the resulted image, and in 4b to 4e are CGs at selected image points. The apparent RMOs in these CGs are evidence of existing velocity errors. The CGs collected from 31 evenly distributed image points are used as data. The sensitivity kernels are calculated for these image points. The dip angles are estimated from the image and reflection angles are from kernel calculations using the method of Xie and Wu (). inally, we use the least-squares method of Lawson and Hanson (1974) to solve the linear system 18. model. The result is shown in ig. 6a and the CGs are in 6b-6e. Comparing to ig. 4, both the image and the CGs are flattened. As a comparison, shown in ig. 7 is the migrated result using the true velocity model. The CGs in igs. 6 and 7 are very similar, indicating the velocity model is considerably improved through the updating process. Conclusions and iscussions The finite-frequency sensitivity kernel is derived for the CGs indexed by surface plane-wave parameters. ased on this sensitivity kernel, a novel wave-equation migration velocity analysis approach is proposed for surface planewave source PM. Numerical model demonstrates the potential application of this method. ince the kernel is formulated for surface plane-wave CG, no expensive local angle or offset decomposition is required. The current numerical test is close to the linear condition. f the initial model is considerably biased from the object model, the nonlinear effect will become apparent and iterations should be required. The current numerical example is for a model. However, there is no major obstacle to generalize this method to 3 problems. igure 6: imilar to ig. 4, except the result is obtained from the updated velocity model. or simplicity, we limit the inversion in an km square with the rest of the model unperturbed. A 6 6 grid is used to control the inversion. Within the grid, a bi-linear function is used to interpolate the model. or details of the model partitioning and kernel storing, see Xie and Yang (8b). hown in ig. 5 is the inverted velocity perturbation which is very close to the original velocity perturbation in ig. 3 and is used to correct the initial model. The migration is then conducted in the updated igure 7: imilar to ig. 4, except the result is obtained from the true velocity model. Acknowledgments This research is supported by the WTOP Research Consortium at the niversity of California, anta Cruz. One of us, Hui Yang, wishes to thank the support from LandOcean Energy ervices Co., Ltd., during the preparation of this manuscript. 3673

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