superposing selected local angle image gathers to obtain improved target images. ... m mn m n z xg x u z zx. G xu z zxu. ,,. ,. ,. ,,. ,. ,. (6) where n m m n z m n j l n m m n z jl mn .... improved target images for different structures (e.g. different dips).
Target-Oriented Prestack Beamlet Migration Using Gabor-Daubechies Frames Ling Chen*, Ru-Shan Wu, Modeling and Imaging Lab, IGPP, University of California, Santa Cruz, CA, USA Yong Chen, China Seismological Bureau, Beijing, P.R.China Summary Target-oriented Gabor-Daubechies beamlet prestack migration is performed with point sources and G-D framebased beam sources for detailed study of the target area and enhancement of certain structural features. The proposed migration method bears considerable advantages in either noise depression or directivity-involved imaging. 2D SEGEAGE salt model prestack data are tested. The obtained results show the improvement of image quality for subsalt structures, especially the steep faults, by selected sources and controlled illuminations. This demonstrates the great potential of the applications of beamlet migration to targetoriented seismic imaging. Introduction The prestack depth migration involves a huge amount of computations required for the extrapolation process of the prestack data, particularly for 3-D surveys. Usually, for the model to be imaged, only a few specific structures are of great interest. These make target-oriented migration with reasonable computational expenses an attracting topic for current research. Moreover, the main principles and processes used for target-oriented purpose may allow detailed analysis on the various possible factors affecting the image quality of the target concerned, and may also bear some advantages for noise depression. To this consideration, Rietveld & Berkhout (1992; 1994) proposed an efficient as well as accurate method to migrate prestack data in a target-oriented way by means of controlled illumination. Prestack depth migration with controlled illumination for the target is a special case of the more general approach of “areal” shot-record migration which was introduced by Berkhout (1992). Although the controlled illumination migration seems superior to the conventional plane-wave stacking method (Taner, 1976) in the sense that the control of the source wavefield is put at the target, the synthesis operator at the surface may become complicated, even may not be obtained for the complex target structures. The plane-wave stacking or plane-wave source migration can be handled much easily at the surface. However, the global plane-wave sources can only control the illumination directivity but lose spatial localization. Furthermore, due to the finite receiver aperture approximations are made in the construction of the recorded fields at the surface generated by plane-wave sources. In fact, construction of record data at the surface corresponding to any type of “areal” sources synthesized from point shots will suffer such kind of
approximations, which becomes more serious in case of global sources, such as plane-waves. Recently, Wu et al. (Wu et al., 2000; Wu & Chen, 2001) developed a beamlet migration scheme with Gabor-Daubechies frame decomposition. They decompose and propagate the wavefield in the G-D beamlet domain. Beamlet decomposition bears both space and direction localization properties, which makes G-D beamlet migration suitable for target-oriented imaging. In this paper, we perform target-oriented G-D beamlet prestack migration with two kinds of consideration: one is from illumination point of view, employing targetilluminating sources (point shots or beam sources) to do partial migration. The other is from structure point of view, superposing selected local angle image gathers to obtain improved target images. The latter is more computational expensive than the former, but can provide directional information of wavefields and structures. Numerical tests on the SEG-EAGE salt model 2D prestack data demonstrate the feasibility and great potential of beamlet migration applied to target-oriented seismic imaging. G-D beamlet wavefield extrapolation With the Gabor-Daubechies frame atoms: g mn x e
im'[ x
g x n' x e i[m x g x xn
(1)
where xn n' x , [ m m' [ with 'x'[ < 2S are the nth window location and the mth local wavenumber, respectively, and g(x) is a Gaussian window function, G-D beamlet decomposition of wavefield at depth z can be expressed as (Wu et al., 2000; 2001): (2) u x, z , Z
¦ ¦ u x , [ z
m
n
m
, Z g mn
n
with the beamlet decomposition coefficients calculated as: u z x n , [ m , Z u x, z, Z , g~mn (3) i[ x ~ dxu x z Z e g x x , , n ³ m
where g~mn x
e i[ m g~ x x n are the dual frame atoms.
Let Gmn represents the Green’s function of gmn propagating from z to z+'z (4) G mn x , z 'z , Z e r iA 'z g mn x , z Here, An is the square-root operators (pseudo-differential operators) n
>
@
An { w 2x k02 xn , z k 2 x, z k02 xn , z
(5)
w 2x k02 xn , z kd2 x, xn , z
in which k0 is the local background (reference) wavenumber (in the nth window), and kd is the local
SEG Int'l Exposition and 72nd Annual Meeting * Salt Lake City, Utah * October 6-11, 2002
Target-Oriented Prestack G-D Beamlet Migration
perturbations of the same window. The wave field at z+'z then can be obtained as the superposition of contributions from all the beamlets: u x, z 'z, Z ¦¦ u z x n , [ m , Z G mn x, z 'z, Z (6) n
m
¦¦ u l
where
u z 'z x l , [ j , Z
¦¦ n
z 'z
x , [
, Z g jl x
G mn , g~ jl u z x n , [ m , Z l
j
m
¦¦ 5 x , [ n
l
j
j
(7)
for deep structures. Fig.1b gives the prestack image using 371 G-D beam sources with small local wavenumbers (propagation angles up to 60q). Its image quality is almost same as that obtained from common-shot migration (Fig.1a). For both cases and the following computations, the G-D frame with 192 windows and 8 local wavenumbers per window is used in wavefield extrapolation. (a)
; x n , [ m , Z u z x n , [ m , Z
m
5 xl ,[ j ; xn , [ m , Z is the wavefield propagator in beamlet
domain. Adopting the local perturbation scheme of decomposing the square-root operator An into background and perturbation parts, the beamlet propagator can be further decomposed into a free-propagator for the local homogeneous media and a perturbation operator to account for the phase-corrections by local perturbations (for details see Wu and Chen, 2001).
(b)
Target-oriented G-D beamlet prestack migration for 2D SEG-EAGE salt model Full beamlet prestack with two types of sources Based on the above G-D beamlet wavefield extrapolation procedure, beamlet prestack migration is performed for the 2D SEGEAGE salt model. The prestack data of this model were generated by 325 shots (NS = 325) with 176 receivers (NR = 176) for each shot. Two types of sources are used here for prestack migration: one is the natural point sources (individual shots), the other is G-D frame-based beam sources. In the common shot migration, no approximation is made for the record data. For the G-D frame-based beam sources, the sources and the data are reorganized by a G-D frame. That is, each beam source is a G-D frame atom: i[ x (8) u Spq x ,0, Z f Z g pq x f Z g x x q e p where f(Z) is the spectrum of the source time function. The beam source u Spq is localized around the space point ( x q, 0) and the horizontal wavenumber [ p . The record data, in this case, need to be resorted to common receiver gather and then synthesized accordingly: (9) u R x ,0, Z u x , x ,0 g x pq
r
¦
r
S
pq
S
xS
Different with the individual shot case, approximations are involved in the construction of the data records. Fig.1a shows the common-shot G-D beamlet prestack image using all the 325 shots. To construct beam sources, the G-D frame with 21 windows and 64 local wavenumbers in each window is used. The total beam source number NB = 21u64 = 1344, much larger than the shot number NS. In fact, beam sources with large local wavenumbers (wide propagation angles) have little impact on the image quality, especially
Fig.1 Images by full G-D beamlet prestack migration for the 2D SEG-EAGE model. (a) using 325 shots; (b) using 371 G-D beam sources Target illumination-based partial source migration As for the SEG-EAGE model, the subsalt structures, especially the steep faults, are mainly concerned for various purposes. Taking the three subsalt steep faults as the specific target, we can see that the target can only be illuminated by the right side of shots or beam sources at the surface. In the common-shot migration, we use the rightmost 125 shots (No. 201-325) to obtain the target image (Fig.2a), while for beam source migration, beam sources in the rightmost 7 windows and with left-propagating angles ranging 15q50q are used, the final image is shown in Fig.2c. In this case, the number of beam sources varies under different frequencies to keep the propagating angles within the same range. For comparison, we give the subsalt residual images in Fig.2b (by subtracting Fig.2a from the subsalt part of Fig.1a) and Fig.2d (by subtracting Fig. 2c from the subsalt part of Fig.1b), respectively. From Fig.1 and Fig.2, it can be obviously seen that the target of subsalt steep faults can be fully imaged by the selected shots or beam sources with reduced background noises. A point source has the highest spatial localization but with no directivity feature, radiating energy into all directions; while a beam source is localized in both space and direction, which is more flexible for directivity-involved imaging. As shown in Fig.2c, for the steep faults, only left-
SEG Int'l Exposition and 72nd Annual Meeting * Salt Lake City, Utah * October 6-11, 2002
Target-Oriented Prestack G-D Beamlet Migration
propagated beam sources (15q50q) are used to process migration, which gives same image quality for the target compared with that resulted from full-directional point sources (Fig.2a). (a)
(b)
(c)
(d)
matrix (or local plane-wave image matrix superposed over frequency, see Wu and Chen, in this volume) is calculated: (11) S R*
IMG x, z, [ m , [ j
¦ u x, z, [
m
,Z u
x, z, [
j
,Z
Z
Local angle image matrix IMG x, z, [ m , [ j can be obtained either by full prestack migration or by partial source migration for target-oriented purpose. It measures the contributions from different directional source – reflection field pairs to the final image. We can output images of different angle pairs (incident angle – scattering angle) as a image album. Fig.3 shows the image album for some angle pairs ( [ m , [ j ) using the same beam sources as in Fig.2c for the subsalt structures. Here [0 represents the zero horizontal wavenumber, [k with k0 the right-propagated horizontal wavenumbers. From Fig.3, we can see that the images with different ( [ m , [ j ) pairs reveal different
Fig.2 Subsalt images. (a) target-illuminating point source migration result (shot No. 201-325); (b) residual image of (a) (by subtracting (a) from the subsalt part of Fig.1a); (c) target-illuminating G-D beam source migration result (beam sources within window 15-21 with propagating angle 15q50q); (d) residual image of (c) (by subtracting (c) from the subsalt part of Fig.1b) Local angle image gathers Above partial source migration is from target illumination point of view to control the source fields at the surface. In this section, by full use of the directivity characteristic of G-D beamlets, we construct the local angle image gathers to extract the directional information and superpose selected angle image gathers according to the target structure for the purpose of further improvement of image quality. In G-D beamlet prestack migration, the source field as well as the reflection field at each extrapolation step (each depth) are obtained as the superposition of contributions from all the G-D beamlets as expressed by (2) or (6). Only superposing the beamlets with same local wavenumbers, we get the source and reflection fields for each individual wavenumber: u S x, z, [ m , Z ¦ u zS x n , [ m , Z g mn x, z (10)
u R x, z, [ j , Z
n
¦ u x , [ R z
l
j
, Z g jl x, z
l
where u zS x n ,[ m , Z and u zR xl , [ j , Z are beamlet coefficients of the source and reflection fields at the image point, respectively. The so obtained u S x, z, [ m , Z and u R x, z, [ j , Z are called the directional source and reflection wavefields (local plane waves). Substituting u S x, z, [ m , Z and
u R x, z , [ j , Z instead of the total source and total reflection
fields to the imaging condition, the local angle image
directivity features. The three subsalt steep faults can be seen more clearly on Fig.3c. It is obvious that only vertical or left-propagated source and reflection fields have contributions to the image of the subsalt structures, especially the target of steep faults. Based on this observation, we superpose the local angle images of only the contributing angle pairs (Fig.3a-d), resulting in noticeable improvement on the image of the steep faults as shown in Fig.4c. For comparison, Fig.4a and Fig.4b give the enlarged subsalt images of Fig.1b and Fig.2c, respectively. From the three figures in Fig.4, we can see that both target-illuminating beam source migration and structure-based superposition of local angle image gathers provide improved image qualities for the target by enhancing the effective signals and suppressing background noises. Using individual shots instead of beam sources to construct local angle image gathers or album can give similar results. Conclusions Target-oriented Gabor-Daubechies prestack beamlet migration can be conducted by the selection of illuminating sources or/and the local angle image matrix elements before stacking. The sources at the surface can be point sources or G-D frame-based beam sources. Employing target-illuminating sources to do partial prestack migration not only saves computation time, but also can suppress noises. Beamlet decomposition and extrapolation of wavefields provide directional information in each migration step, from which local angle image matrix can be constructed to monitor the contributions from different directional incident-reflection field pairs to the final image. Superposition of selected local angle images gives improved target images for different structures (e.g. different dips). Numerical tests on the 2D SEG-EAGE salt
SEG Int'l Exposition and 72nd Annual Meeting * Salt Lake City, Utah * October 6-11, 2002
Target-Oriented Prestack G-D Beamlet Migration
model prestack data show the feasibility and great potential of G-D beamlet migration in target-oriented and structureoriented imagings. (a)
(a)
(b)
(b) (c)
(d)
(e)
(f)
(g)
(h)
(c)
Fig.4 Subsalt images. (a) enlarged from Fig.1b; (b) enlarged from Fig.2c; (c) result by superposing local angle images with angle pairs ( [0 , [0 ), ( [0 , [1 ), ( [1 , [0 ) and ( [1 , [1 )
Fig.3 Local angle-image album for the subsalt structure. (a) ( [0 , [0 ); (b) ( [0 , [1 ); (c) ( [1 , [0 ); (d) ( [1 , [1 ); (e) ( [0 , [1 ); (f) ( [1 , [1 ); (g) ( [0 , [2 ); (h) ( [1 , [2 ) References Berkhout, A.J., 1992, Areal shot record technology, J. Seis. Expl., 1(3), 251-264. Rietveld, W.E.A., Berkhout, A.J. and Wapenaar, C.P.A., 1992, Optimum seismic illumination of hydrocarbon reservoirs, Geophysics, 57(10), 1334-1345. Rietveld, W.E.A. and Berkhout, A.J, 1994, Prestack depth migration by means of controlled illumination, Geophysics, 59(5), 801-809.
Taner, M.T., 1976, Simplan: similated plane-wave exploration, 46th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 186-187. Wu, R.S. and Chen, L, 2001, Beamlet migration using Gabor-Daubechies frame propagator, 63rd Conference & Technical Exhibition, EAGE, Expanded abstracts, 74. Wu, R.S., Wang, Y. and Gao, J.H., 2000, Beamlet migration based on local perturbation theory, 70th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded abstracts, 1008-1011. Acknowledgements The supports from the WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project and the DOE/BES Project at University of California, Santa Cruz are acknowledged.
SEG Int'l Exposition and 72nd Annual Meeting * Salt Lake City, Utah * October 6-11, 2002
