Prestack Migration/Imaging Using Synthetic Beam-Sources and Plane-Sources

Transcription

1 Pretack Migration/Imaging Uing Synthetic Beam-Source and Plane-Source Ru-Shan Wu, Ling Chen and Yongzhong Wang Modeling and Imaging Laboratory, IGPP, Univerity of California, Santa Cruz, CA 95064, USA Abtract The principle and procedure of beam-ource ynthei and beam-ource migration for wave-equation baed imaging method are introduced and teted with Gabor-Daubechie and local-coine pretack beamlet migration. Plane-ource ynthei and migration are treated a a pecial cae of approximation to the beam-ource ynthei and migration. Beam-ource with properly elected window provide both pace and direction localization for illumination and i very flexible for target-oriented migration uing wave-theory baed method. On the other hand, plane-ource migration can increae the computation efficiency of depth migration ignificantly, up to 5-10 time for the SEG-EAGE 2D alt model and Marmoui model. Introduction Reflection eimic data are collected in hot gather. Each gather i a point ource-excited wavefield recorded by a receiver array on the urface. Such a data et can be rearranged (reorted) for different purpoe, uch a the CMP gather for Kirchhoff migration. Wave-equation baed imaging method, epecially FD method, uually are implemented in hot domain. The imaging proce i to downward extrapolate the point ource field and the received reflection field imultaneouly, and then apply the imaging condition in the image pace. For the offhore data acquiition, each hot ha a much limited receiver aperture, but the hot number could be tremendou. However, the downward extrapolation of the ource field need a rather large aperture to include all the wave path. Therefore the hot-domain pretack depth migration i time-conuming and inefficient. Plane-ource have been propoed and applied to pretack depth migration, ee [1], [3], [8-10], [12] and [17], to peed up the computation. Beam-ource alo have been introduced and employed in the pat [11] and [13]. However, in Tygel et al. paper, only the vertically incident beam are conidered. Sun et al., propoed to group the eimic trace in a way uch that the ource and receiver of the trace fall into mall urface patche. Then the grouped trace can be migrated together by Kirchhoff migration, correponding to migration uing both ource and receiver ray beam. In thi paper, we propoe to rearrange and yntheize the data into beam-ource excited data for waveequation baed imaging method. Beam-ource have controlled beam width, beam location and beam direction, and therefore can be applied to target-oriented and tructure-oriented imaging in an efficient way [2]. The beam-ource can be further combined and approximated to plane-ource, reulting in great improvement in computation efficiency for depth migration. The migration method ued here i the beamlet migration with Gabor-Daubechie (G-D) frame propagator or local-coine bai (LCB) propagator [15-16]. However, the principle and method of beam-ource and plane-ource ynthei dicued here are applicable to other wave-equation baed method a well.

2 Synthetic beam-ource Gauian beam method ha been introduced into eimology for calculating ynthetic eimogram in heterogeneou media [4, 6]. In wave propagation imulation, the wavefield of a point ource or a plane ource on the urface can be decompoed into Gauian beam [4]. Propagation of individual Gauian beam ha been tudied extenively by high-frequency aymptotic theory, e.g. in [5-7]. In thi paper, we will ue the beam decompoition and propagation for a different purpoe. The technique of ynthetic aperture and beamforming are ued to tranform a collection of point-ource eimogram into a collection of beam-ource eimogram. The beam-ource field and the correponding eimogram are extrapolated to the image pace by wave-equation baed method, here the beamlet propagation method developed in [15-16]. The hot-gather eimogram are reorted and yntheized into beam-gather eimogram, which correpond to a data collection proce uing beam-ource. Firt the hot gather record u(x g, x ) are reorted into common receiver-gather record u(x, x g ): x = x 1,, x N Then beam-ource ynthei i performed to each receiver gather: * ( x, k, x ) u( x, x ) b ( x, x k ) u g = g, A where * tand for complex conjugate, and A i the ource aperture for the receiver at x g, b ( x, x, k ) i a beam function: b ( ) ( ) ik x x, k = g x x e x dx, (2) with g(x) a the window function, which determine the beam hape, x i the location of beam center and k i the local wavenumber which pecifie the beam direction. The beam function b ( x x, k ), can be an orthogonal bai (e.g. local coine bai) or a over-complete bai (e.g. Gabor-Daubechie frame). In the latter cae, the original point ource-excited data can be totally recovered with the recontruction formula ~ u x, x u x, k, x b x, x, k (3) where ( g ) = ( g ) ( ) ~ b k x ( ) ( ) ik x, x, k = g ~ x x e x i the recontruction bai with g ~ ( x) a the dual window function. In the cae of orthogonal bai, g ~ = g, o b ~ = b. In modern multi-fold eimic data acquiition, the information contained in the data i nonuniformly ditributed for image recontruction. It may have much redundancy for ome part, but have very little coverage for other part of the image pace. Therefore, the tacking of the yntheized ource i not to totally recover the original excitation configuration, but to enhance the effective illumination of the target area for quality, and reduce redundancy in other area for efficiency. Thu the manipulation and tacking of the ynthetic beam-ource i the art of target imaging, but the technique of beam-ource ynthei provide the bai and flexibility for the artitry. If we take the window function a a rectangular window covering the full aperture A, then the beamource data in (1) become (1) (4)

3 ( x k, ) = u( x ) e ik x, dx (5) A u, which i equivalent to the beam-forming proce in plane-ource migration ued in [1], [3], [8-9], and [17]. It i a pecial cae of ynthetic beam-ource. Thi kind of beam ha much wider width o that it can be approximately conidered a plane wave in cae of large A. However, ince it adopt the rectangular window, there are ide lobe (edge diffraction) of the beam propagation which could bring evere ditortion to the wavefield of the plane wave, epecially near the beam edge at hallow depth and in complex region. Fig.1 compare the propagation of rectangular-window beam-ource with thoe of Gauian-window beam-ource (G-D beam-ource) in the homogeneou medium and Fig.2 give the imilar comparion for the SEG-EAGE alt model. We can ee clearly the edge effect of the rectangular beam. Propagation of local-coine beam-ource in the homogeneou medium are hown in Fig.3. Different from the G-D beam-ource cae, the wavefield of one local-coine beam-ource correpond to two ymmetric beam a can be een clearly in Fig.3b and 3c. For the vertically incident beam-ource, the two ymmetric beam merge into one beam with wider width (Fig.3a). (a) (c) (d) Fig.1 Comparion of wavefield excited by Gauian-window beam-ource (G-D beam-ource) and by rectangular-window beam-ource in homogeneou medium. On the left are wavefield of G-D beam-ource: (a) vertical incidence; oblique incidence; On the right are wavefield of rectangularwindow beam-ource: (c) veritical incidence; (d) oblique incidence.

4 (a) (c) (d) Fig.2 Comparion of wavefield excited by Gauian-window beam-ource (G-D beam-ource) and by rectangular-window beam-ource for the SEG-EAGE alt model. On the left are wavefield of G-D beam-ource: (a) vertical incidence; oblique incidence; On the right are wavefield of rectangularwindow beam-ource: (c) veritical incidence; (d) oblique incidence. (a) (c) Fig.3 Wavefield excited by local-coine beam-ource in homogeneou medium. (a) vertical incidence; mall-angle oblique incidence; (c) large-angle oblique incidence.

5 Plane-ource approximation for beam-ource excitation The pretack depth migration uing beam-ource data and beam-ource excitation involve downward extrapolation of both the beam-ource and the data to the image pace and apply the image condition there. Stacking all or part of the beam-ource image depending on the tak of the imaging, the final image can be obtained. Since each beam-ource ha to be downward propagated eparately, the proce i time conuming. We will try to increae the efficiency by replacing individual beam-ource excitation by global plane-ource excitation. If we put a unit-trength beam-ource I ( x, k ) on the urface, at any depth the incident field can be obtained a G( x k, z) function. Although the amplitude of G( x k z),,, where G tand for the beam-ource Green, will vary along the horizontal direction, but the phae front will be nearly ame a the plane-wave excitation. Therefore, we can apply a plane-ource excitation to approximate the incident field of individual beam ource. By thi way, the computation time for ource field extrapolation can be aved coniderably. Plane-ource migration/imaging If we tack all the beam-gather data of the ame wavenumber u ( x k, x ) aperture x ( x k, ) = u( x ) g( x x ) A, over the whole ource ikx, e dx (6) x u, and choice the window function g(x) atifying Then after normalization with the contant, we obtain x g ( x x ) = cont, x A ( k ) = u( x, k, ) = u( x ) ik x, e dx (7) A x u, Thi i the beam-forming or Fourier tranform for the ynthei of global plane-ource excitation. In thi way the data are tranformed into plane-ource gather, o that plane-ource migration can be conducted with much increaed efficiency. The number of k for imaging depend on the ource aperture A. If A keep contant for all the receiver, we can take k = 2π / A. We limit the maximum k to k m = ω / v with v the average velocity on the urface, o the number of plane-ource i proportional to the frequency. For deep target, we can alo limit the larget angle of plane wave to reduce the computation. From our experience with 2D beamlet migration uing local coine bae or G-D atom, 5 10 time of reduction in computation can be achieved. We know that the reorted data excited by plane-ource are, in fact, till beam-ource gather with rectangular wide window due to the ource aperture limitation for each receiver. Uing global planeource excitation to replace beam-ource excitation i an approximation. The phae relation will be accurate, but the amplitude ditribution of the two type of excitation may be different. Thi approximation will influence the pretack image quality to ome degree. Plane-ource migration i ued to achieve high computation efficiency. We tet the SEG-EAGE alt data with the local-coine beamlet migration and the Marmoui data with the G-D beamlet migration uing plane-ource. The obtained image are hown in Fig.4b and Fig.5b, repectively. In comparion, Fig.4a g

6 and Fig.5a give the correponding common-hot beamlet pretack image (from [14] and [2], repectively). For the SEG-EAGE alt model, we ue 53 plane-ource (the correponding ray parameter p ranging from -520 to 520 µ/m). For the Marmoui model, the number of plane-ource i frequencydependent. For the highet frequency, 73 plane-ource are ued. Averaging over all the frequencie, 46 plane ource are ued per frequency. For both model, plane-ource illumination angle varie approximately from 50 to 50 on the urface. We can ee that the overall image qualitie of both cae are comparable with the common-hot migration image. Taking only the vertically incident plane-ource for migration, the image of the two model are hown in Fig.4c and Fig.5c, repectively. Mot of the horizontal or near horizontal tructure of the model are imaged correctly by the vertically incident plane-ource. (a) (c) Fig.4 Comparion of plane-ource and point-ource imaging in pretack depth migration for the SEG- EAGE alt model uing local-coine beamlet. (a) point-ource (common-hot) image with 325 hot; plane-ource image with 53 plane-ource; (c) ingle plane-ource image (vertically incident).

7 (a) (c) Fig.5 Comparion of plane-ource and point-ource imaging in pretack depth migration for Marmoui model uing G-D beamlet. (a) point-ource (common-hot) image with 240 hot; plane-ource image (89 plane-ource for the highet frequency); (c) ingle plane-ource image (vertically incident). Beam-ource migration/imaging Synthetic beam ource i the mot flexible type of ource for target-oriented imaging. The harpne of the beam directivity can be adjuted by changing the initial beam width. After beam-ource ynthei of the data, imilar procedure of double downward extrapolation for both the ource field and the reflection wavefield are performed for beam-ource migration/imaging. Different from plane-ource cae, we can control not only the propagation angle but alo the pace location of the beam-ource to

8 meet certain migration requirement. We apply the beam-ource migration to the SEG-EAGE alt model pretack data to tet the feaibility of beam-ource migration. Fig.6 how two example of ingle Gauian-window beam-ource image with G-D beamlet migration. In the figure, the outline of model tructure are alo plotted. We ee that different beam can illuminate and recontruct different part of the alt-body and different part of the ubalt tructure. To contruct the overall image for the whole pace, the image of different beam-ource at different location and with different propagation direction are uperpoed together. However, in practical computation, beam-ource with large local wavenumber (wide propagation angle) have little impact on the image quality, epecially for deep tructure. Fig. 7 how the beam-ource pretack depth migration for the SEG-EAGE alt model. Fig 7(a) i the image uing 162 local-coine beam-ource and Fig. 7 i that uing 371 Gauian-window beam-ource. In both cae, the propagation angle of beam-ource are limited within on the urface. We ee that the image qualitie are comparable with the hot-domain pretack image. (a) Fig.6 Single beam-ource image of the SEG-EAGE alt model by different Gauian-window beamource (G-D beam-ource).

9 (a) Fig.7 Pretack beam-ource migration image for the SEG-EAGE alt model. (a) uing 162 local-coine beam-ource; uing 371 G-D beam-ource. Application to target-oriented imaging Synthetic ource, epecially beam-ource, are flexible for imaging in a target-oriented way to achieve good image quality [2]. A for the SEG-EAGE alt model, the ubalt tructure, epecially the teep fault, are mainly concerned for variou purpoe. Here we take the three ubalt teep fault a the pecific target of migration. From target-illumination point of view, it i obviou that only right-ide located ource on the urface have contribution to illuminate the targeted teep fault. For point-ource (common-hot) migration, we elect the rightmot 125 hot (hot No ) to obtain the target image a hown in Fig.8b. Uing a coine-tapered window covering the ame ource aperture of the 125 hot, we contruct a kind of beam-ource with much wider beam width. Fig.8c give the ubalt image by thee wide beam-ource with propagation angle ranging In Fig.8d, G-D beam-ource in the rightmot 7 window (window No ) and with left-propagation angle ranging are ued to obtain the final image. For comparion, the full common-hot migration ubalt image i given in Fig.8a. We can ee that by uing target-illuminating ource, the artifact at the left ide of the ubalt area are uppreed in both Fig.8b, 8c and 8d. Effective ignal of ubalt teep fault are further

10 enhanced through collimated illumination of the beam-ource. The teep fault in the middle and the left are hown more clearly than the cae of point-ource illumination. (a) (c) (d) Fig.8 Subalt image by pretack G-D beamlet migration. (a) point-ource (common-hot) migration image uing all the 325 hot; target-illuminating point-ource migration image (hot No ); (c) target-illuminating coine-tapered rectangular beam-ource migration image (with the ame ource aperture a in and left-propagation angle ) (d) target-illuminating G-D beam ource migration image (beam ource within window with left-propagation angle ) Concluion Synthetic beam-ource have localization in both pace and direction. Beam-ource migration i very flexible for target-oriented imaging uing wave theory-baed method. On the other hand, plane-ource migration can increae the computation efficiency of depth migration ignificantly, up to 5-10 time for the SEG-EAGE alt model and Marmoui model. Acknowledgement The upport from the WTOPI (Wavelet Tranform On Propagation and Imaging for eimic exploration) Reearch Conortium at Univerity of California, Santa Cruz i acknowledged.

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