Dreamlet source-receiver prestack depth migration

Transcription

1 Dreamlet source-receiver prestack depth migration Bangyu Wu 1, Ru-shan Wu 2 and Jinghuai Gao 3 1 Xi an Jiaotong University, Department of Electronic and Information Engineering, Institute of Wave and Information, Xi an, Shaanxi, China and University of California, Santa Cruz, Modeling and Imaging Laboratory, Institute of Geophysics and Planetary Sciences, 1156 High Street, Santa Cruz, California bangyuwu@gmail.com. 2 University of California, Santa Cruz, Modeling and Imaging Laboratory, Institute of Geophysics and Planetary Sciences, 1156 High Street, Santa Cruz, California E- mail: wrs@es.ucsc.edu. 3 Xi an Jiaotong University, Department of Electronic and Information Engineering, Institute of Wave and Information, Xi an, Shaanxi, China jhgao@mail.xjtu.edu.cn. Right Running Head: Dreamlet source-receiver prestack depth migration 1

2 Dreamlet source-receiver prestack depth migration ABSTRACT Dreamlet migration is seeking to develop algorithm for the decomposition and compression of the seismic wavefield and the one-way wave equation operator based on the wavelet transform and local perturbation theory. For the wavefield, dreamlet applies the wavelet transform on both the spatial and temporal coordinates. It adds wavelet transform to the frequency domain beamlet method by an additional wavelet transform along the time axis which achieves good wavefield compression. One-way propagator is a forward marching algorithm, which extrapolates the wave in preferred direction. The dreamlet one-way propagator takes advantage of this feature and migrates the wavefield directly in the compressed domain. During dreamlet depth migration, the seismic data always flows in one direction along the time axis and part of seismic data used to image the shallower structures will flow out the time space panel. The deeper the depth is, the shorter the valid time axis and the less computation for one depth step continuation. In order to fully explore this feature, we combine the dreamlet migration with the idea of survey sinking. Survey sinking downward-continues all the recorded data at once and moves the reflections to the spatial position of the reflecting point. The propagated wavefield is equivalent to the data that would have been recorded if all sources and receivers were placed at the working depth level. The dreamlet survey sinking process only keeps the data used to image the structures beneath the survey system. We use the source-receiver survey sinking algorithm because it can be easily incorporated with the local perturbation theory. In this work, both the spatial and temporal localizations adopt the orthogonal Local Cosine Basis (LCB) transform. This not only constructs an 2

3 orthogonal version of dreamlet but also keeps the wavefield dreamlet coefficients and the one-way dreamlet migration operator in real data type, which means, all the migration and imaging process are completed in the real data domain. A three-scatter model is used to demonstrate the computational feature and principle of this method and the 2D SEG/EAGE salt model is used to show the imaging quality of this method on the complex geology structures and strong velocity contrast model. INTRODUCTION The dual domain (space-wavenumber domains) one-way wave equation based propagators, generally speaking, tend to generate a better migrated image comparing with the frequency-space or frequency-wavenumber domain methods, because best operations can be applied in the corresponding domains in terms of accuracy and efficiency. In dual domain methods, the heterogeneous velocity is decomposed into a background velocity and perturbation that accounts for lateral velocity variations. In the approaches, the background velocity can be global or local. For the global perturbation dual domain methods, such as SSF (Stoffa et al., 1990), FFD (Ristow et al., 1994) and GSP (Wu, 1994, Xie and Wu 1998, Jin et al., 1999, De Hoop et al., 2000), the medium is decomposed at each depth level into just one background medium and perturbations account for global velocity variations. For the treatment of strong contrast velocity, such as the cases involved in subsalt imaging, the global velocity perturbations can be very large, leading to difficulties in correctly propagating high-angle waves. In addition, for super-long profiles, the use of long-line Fast Fourier Transform (FFT) greatly increases the computation intensity. 3

4 In local perturbation theory (Wu et al. 2000), the wavefields are spatially localized with local windows and propagated with local reference velocity which adjusts the lateral velocity variations. Weaker perturbation significantly improves the accuracy of wave propagation in complex media and provides good imaging results when compared to the traditional wave equation based methods. Migration methods along this approach have been developed using the Gabor-Daubechies frame (GDF) (Wu and Chen, 2001, 2002a, 2002b, 2006; Chen and Wu, 2002; Chen et al., 2006) and the Local Cosine Basis (LCB) (Wang and Wu, 2002; Luo and Wu, 2003; Wang et al.,2003, 2005; Wu et al., 2003; Wu et al., 2008). ). In these methods, wavefields have been Fourier transformed over time and wavelet transformed over space. Those frequency-space-wavenumber transforms are usually referring to as Beamlet transform. In beamlet migration, the wavefields are spatially localized with uniformly distributed windows and propagated with beamlet propagators (sparse propagator matrices). Generally speaking, selecting window length results in trade-off between the efficiency and accuracy. Short window length is more flexible to adapt to the lateral velocity variations but more expensive because computation costs are proportional to the number of windows. In order to improve the efficiency, Ma and Margrave (2008) developed an adaptive window partitioning scheme relating window length to lateral velocity gradient based on the unity partition Gabor transform and referred as locally homogeneous approximation. Pedersen et al. (2010) shared the similar idea and proposed dividing the velocity medium into windows containing sediments, salt-sediment interface and salt only and use different migration operators in different windows. However, all the above mentioned methods mainly focus on migration operator 4

5 compression and keep the data (wavefield) in frequency domain. One interesting and important approach would be to study both data compression and migration operator compression using wavelet transform, and then process the compressed data directly in the wavelet domain. Seismic data compression is intended to find efficient ways of representing data. The wavelet basis used to decompose the wavefield must be localized on time because the seismic events have very short time duration comparing with the recording time. Basis localized on time can easily fit the seismic events and can achieve better wavefield compression. Recently, curvelet transform (Candès and Donoho., 2002; Candès and Demanet, 2005), which is complete space-time localized frames, has been implemented to seismic compression successfully and has been applied to wave propagation and seismic imaging using a map migration method (Douma and Maarten, 2007; Chauris and Nguyen, 2008). In these methods, high-frequency asymptotic approximation was invoked to propagate the curvelets in smoothly inhomogeneous media. Dreamlet migration method (Wu et al., 2008) extended the frequency domain beamlet to time-space domain wave propagation, by applying multi-dimensional local harmonic transform to the wavefield which is on the transversal plane of the one-way wave propagation direction. We call the temporal localized atom drumbeat, and the spacewavenumber atom beamlet. Dreamlet (combination of the terms drumbeat and beamlet) is generated by the tensor product of drumbeat and beamlet, forming the time-frequencyspace-wavemumber localized atom. Unlike the curvelet transform, which takes the 2 dimensional time space seismogram as picture, in dreamlet, the temporal localization can be treated separately and differently from the spatial localization. In Wu et al. (2008, 5

6 2009), the time localization adopts the local exponential frames (Auscher, 1994; Wu and Mao, 2007; Mao et al., 2007), which have redundancy 2, and the wavefield coefficients and propagator matrix stay in the complex data domain. In order to improve the efficiency, orthogonal dreamlet (Wu et al., 2010) is constructed by applying the LCB on both of the time and space localization. For the wavefield decomposition, the orthogonal dreamlet is similar to the 2 dimensional LCB transform, which is defacto standard in the image compression in image processing industrial area. Orthogonal dreamlet decomposition not only reduces the amount of the wavefield dreamlet coefficients and propagator dataset, but also keeps the data type to real. All the propagation and imaging process is finished in the real data domain. Until now, the dreamlet migration is only implemented in the shot domain. In short profile migration, the source and receiver wavefield are downward-continued independently and cross-correlated to produce the image at each depth. One interesting phenomenon in shot profile dreamlet migration is that, on the receiver side, the valid length of the time axis is becoming shorter as the depth increasing because the dreamlet migration moves the seismogram to one direction along the time axis and some data flows out of the time-space panel as depth increasing. However, the point source excited source wavefield expands during migration on the time-space plane and then increases the amount of dreamlet coefficients and therefore, the computation time. Survey sinking imaging (Claerbout, 1985) bases on the principle of reciprocity and downward continues the entire seismic survey at the same time. The propagated wavefield is equivalent to data that would have been recorded if all sources and receivers were placed at the current depth level (Biondi, 2007). Beamlet survey sinking prestack 6

7 depth migration is investigated in offset domain (Luo et al., 2006). Survey sinking migration by the dreamlet propagator can only keep the data used to image the structures beneath the survey system and get rid of the data which has been used to image upper depth structures automatically. In the dreamlet prestack survey sinking scheme, the LCB transform is applied to the common source, common beamlet receiver and time axis, which is a fully wavefield decomposition. Therefore, the wavefield compression ratio in the survey sinking is higher than the shot domain, which decomposes the source and receiver wavefield separately. In principle, these properties can be major factors to speed up the migration process and still get good image. In this paper, we first describe the concept of the orthogonal dreamlet and present the dreamlet coefficients of the SEG/EAGE model on the surface, before migration. Then we describe the analytical dreamlet one-way propagator. Following that, we show the imaging condition for the shot domain and the survey sinking and demonstrate how the dreamlet migration takes advantage of the one-way approximation. Performance of the method on the high contrast salt model is illustrated on the 2-D SEG/EAGE model. ORTHOGONAL DREAMLET AND PRESTACK WAVEFIELD DECOMPOSITION In 2D case, at a given depth, the data set are defined in a 3D continuum (, s r, ) where s x and u t x x, r x stands for the source and receiver location and t the recording time. It can be decomposed by local Fourier frames, such as the Gabor-Daubechies Frame, or the local trigonometric basis (local cosine/sine basis). However, it was proved that orthogonal local Fourier basis cannot exist. Historically, it was discovered independently by two physicists, R. Balian (1981) and F. Low (1984). Later G. Battle (1988) (can t find 7

8 reference) proved this theory from the Heisenberg uncertainty principle. As a result of this theorem, if the wavelet is an orthogonal basis, then the Heisenberg product, i.e. the time-frequency resolution-cell (uncertainty), must be infinite. In the case of Gabor s critical sampling, the frequency spreading is infinite: no frequency localization at all. This seemingly insurmountable obstacle gave a blow to the enthusiasm of the search for orthonormal local Fourier basis, and had blocked the way of finding efficient timefrequency localization method for many years. To circumvent the Balian-Low obstruction, physicist K. Wilson (Nobel laureate) (Wilson, 1971) (can t find reference), when working in quantum mechanics, proposed to replace the exponential function with the cosine and sine functions alternatively. Later H. Malvar (1990), a scientist in signal processing, discovered a similar approach independently. Significant improvement along this line has been made (Daubechies et al., 1990, 1992; Coifman and Meyer, 1991; Wickerhauser, 1994) and the local cosine/sine bases with adaptive window lengths were constructed. One upper hand of the local cosine/sine transform is that it does not change signal data type and this is why, after decomposition, the orthogonal dreamlet coefficients stay in real data domain. Local Cosine Basis (LCB) and orthogonal dreamlet basis LCB constructed by Coiffman and Meyer (1991) uses overlapped bell functions to window the cosine oscillation and is developed as a type of orthogonal wavelet basis (see also Mallat, 1999). In the space domain, the LCB element can be characterized by position x n, the interval (the nominal length of the window) Ln xn+ 1 xn =, and wavenumber index m ( m = 0 M-1, M denotes the total sample points of the interval) as follows 8

9 2 bmn ( x) = Bn ( x) cos( ξm ( x xn )), (1) L n Where ξ m π 1 = m + L 2 n and Bn () x is a bell function which is smooth and supported in the compact interval [ xn ε, xn+ 1 + ε '] for xn ε xn+ 1 + ε ', where ε, ε ' as the left and right overlapping radius. Similarly in time domain, LCB can be characterized by time location t j, time interval T = t t, and local frequency index i as follows j j+ 1 j π 1 andωi = i + T 2. j 2 () () cos( ω ( )) b t = B t t t, (2) ij j i j Tj In shot domain migration, the orthogonal dreamlet basis can be generated by the tensor products of bmn ( x ) and b () t, ij ( ) ( ) ( ) ( ) 2 2 () ( ) t,,, ( ) j i xn ij mn j n m ( i j ) ( m ( n) ) d t, x = b t b x = B t B x cos ω t t cos ξ x x (3) ω ξ T L j n The nominal supports of the dreamlet basis are the Cartesian product rectangles of the nominal supports of the time and space and decompose the source and receiver wavefield separately. Source-receiver migration is based on the concept of survey sinking. Survey sinking imaging downward continues the sources as well as the receivers. Figure 1 is a schematic illustration about resorting the shot-gather seismograms and computing aperture for the 2D SEG/EAGE salt model. We use a three-dimensional LCB decomposition to transform the dataset from point source, point receiver and time record into a beamlet-source 9

10 s r b ( x ), beamlet-receiver b ( x) and drumbeat b () t (time-frequency localized atom) mn pq records. That means the dreamlet basis used for source-receiver survey sinking imaging, s r in the 2D case, is generated by tensor products of b ( x ), b ( x ) and b () t, d t x x b t b x b x sr s r s s r r (,, ) = ij ( ) mn ( ) pq ( ) ( t j, ωi, xn, ξm, xq, ξ p) s r s r Bj() t Bn( x ) Bq( x ) cos( ωi( t t j) ) cos( ξ m( x xn) ) cos( ξ p( x xq) ) = T L L j n q π 1 whereξ p = p + L 2. q The observed data (, s r, ) ij mn pq ij, (4) u t x x at depth z is decomposed into dreamlets with windows along the common source, common receiver and time axis, ( ) = ( ) s r s r sr s r sr s r u t, x, x u t, x, x, d ( t, x, x ) d ( t, x, x ) = m n p q i j m n p q i j ( t,,,,, ) j i x n m x q p ( j ωi n ξ m q ξ p) sr s r u t,, x,, x, d ( t, x, x ) μ μ μ, (5) μ = ω ξ ξ is the local parameter cluster,, stands for inner product with (, s, r ), sr (, s, r ) = s r (, s, r ) sr (, s, r ). u t x x d t x x dtdx dx u t x x d t x x μ Figure 2 shows some common beamlet sources and receivers dreamlet coefficients for the 2D SEG/EAGE salt model. The spatial and temporal windows are uniformly partitioned and nominal window lengths are 16 sampling points respectively. Those figures show the sparsity in the transformed domain. From those figures, we see that most of the energy is in the low wavenumbers and frequencies because of the energy compaction property of the LCB transform. During migration, all the coefficients in the same space window are migrated to next depth based on the local reference velocity in the corresponding window. μ 10

11 We only migrate the large coefficients and ignore the small ones and there is trade-off between the efficiency and accuracy for different strategy on how to threshold the transformed data, however, this topic is beyond the study of this paper. In the following applications, we set the threshold as depth of the maximum absolute coefficients at every SOURCE-RECEIVER SURVEY SINKING IMAING IN DREAMLET DOMAIN Source-receiver migration is dependent on downward continuation of the whole recorded seismic data and based on a recursive solution of the one-way or full way wave equation (Sandberg and beylkin, 2009, Alkhalifah and fomel, 2010). Here, the dreamlet sourcereceiver imaging is based on the one-way wave equation. Source-receiver survey sinking dreamlet propgator Starting with Claerbout s (1985) source-receiver Double Square Root (DSR) equation for a laterally varying velocity medium in time-space domain: 1 t 2 1 t 2 u t x x u t x x 2 s s 2 r r z v ( x ) x v ( x ) x t s r s r (,, ) = + (,, ) s r where vx ( ) and vx ( ) location correspond to the velocities at the source location, (6) s x and receiver r x at depth z, respectively. The DSR equation means the sources and receivers are downward continued simultaneously. For the source-receiver survey sinking imaging, the DSR equation (6) can be separated into two Single Square Root (SSR) equations: 1 t u t x x u t x x 2 s s z v ( x ) x t s r s r (,, ) = (,, ) 2, (7) 11

12 1 t u t x x u t x x 2 r r z v ( x ) x t s r s r (,, ) = (,, ) 2. (8) Equation (7) downward continues the receiver field (common source) and the other for the source field (common receiver). These two equations share the same form and commute via the data set (, s r, ) u t x x. This means downward continuing the sources is really the same thing as downward continuing the receivers and can be done sequentially. We can first downward continue the source wavefield (Equation 7) for every receiver and then receiver wavefield for every source (Equation 8). In the following for the formula derivation, we only take equation (8) as example. The solution to equation (8) is and 2 2 r t i x Δz ( ) 2 r r s v ( x ) s r, ; (,, ) u +Δ t x x = e u t x x, (9) z z z S SSR 2 2 r t i x 2 Δz r v ( x ) P = e, (10) is the exact single square root solution. In order to use the local perturbation theory, the wavefield is decomposed into dreamlets with windows along space and time and the exact SSR propagator is separate into a local background propagator S 0 P SSR and local phase-screen corrections ( ) S1 P SSR r s S S sr s r S S sr s r uz+δ z t, x ; x = P P SSR SSR u d (, t x, x ) = P SSR u P SSRd (, t x, x ). (11) (,,,,, ) j i n m q p μ μ μ μ μ μ u = u t ω x ξ x ξ is the simple note of the dreamlet coefficients located at μ source space window x n, receiver window x q, source wavenumber ξ m, receiver 12

13 wavenumber ξ p, time windowt j and frequency ω i. According to the square-root approximation, or the symbol of the pseudo-differential operator (for the approximation see Wu et al., 2000; Chen et al., 2006), the local background propagator is expressed as S 0 SSR 2 2 r t i x 2 0 ( r ) Δz v x p P = e, (12) where ( r v ) 0 x are the local reference velocity in the corresponding windows. p The evolution of dreamlets must observe the wave equation. After propagation, a dreamlet is no longer a dreamlet, and is spreading into other cells in the localized phasespace. The redecomposition of the distorted dreamlet into new dreamlets forms the propagator matrix elements: 1 0 ( ) P P r s S S sr s r sr s r sr s r uz+δ z tx, ; x = SSR u SSRd (, tx, x), d (, tx, x) d (, tx, x) μ μ μ' μ' μ' μ. (13) = P u P d (, tx, x), d (, tx, x) d (, tx) S1 S 0 sr s r sr s r sr r SSR μ SSR μ μ' μ' μ' μ The parameters with prime denote the basis after propagation. So the dreamlet propagator matrix elements used for the survey sinking migration are PSSR ( μ' μ ) = P d ( t SSR, x, x ), d ( t, x, x ), (12) 0S S0 sr s r sr s r μ μ' which represent the dreamlet propagation and the weight of cross-coupling of different dreamlet basis. According to the dreamlet basis (equation 4) in (12), we have 13

14 P = = = ( μ' μ) = P d ( t, x, x ), d ( t, x, x ) 0S S0 sr s r sr s r SSR SSR μ μ' 2 t 1 i Δz 2 r 2 r s r v0 ( xp ) x s s r r s s r r ij mn pq i j m n p q () ( ) ( ) ' '() ' '( ) ' '( ) dtdx dx e b t b x b x b t b x b x 2 t 1 i Δz s s s s s r r r r r mn m n ij pq i j p q 2 2 ( ) ( ) () ( ) () ( ) 0 r r v ( xp ) x ' ' ' ' ' ' 0 ( x ) b ' '( x ) P ( t j', ωi', xq', ξ p' ; t j, ωi, xq, ξ p) dx b x b x dtdx e b t b x b t b x s s s s s S dx bm n m n SSR, (13) 0S where P ( t j', ω i', x q', ξ '; t SSR p j, ω i, x q, ξ p ) is the single square root dreamlet propagator and the definitions of fourier transform from space to wavenumber and time to frequency are iξ x f ( ξ) = dxe f ( x), (14) iωt f ( ω ) = dte f ( t). (15) Because the orthogonality of local cosine basis, equation (13) can be expressed as P AP ( t, ω, x, ξ ; t, ω, x, ξ ), if m= m' and n= n' =, (16) 0. other 0S 0 S SSR j' i' q' p' j i q p SSR ( μ' μ) and A dx b ( x) 2 =. mn From equation (16), we see that the source-receiver survey sinking dreamlet propagator is same as the shot domain. The survey sinking operations now changed from a point source common source gather to a common beamlet source gather. After propagation, the common beamlet source gather does not spread into other common source gathers. In the following, we use the analytic solution of the wave equation in the frequencywavenumber domain in a homogeneous background for the derivation of dreamlet propagator in the background media. 14

15 P 0S SSR ( t, ω, x, ξ ; t, ω, x, ξ ) = j' i' q' p' j i q 1 r * * r r real dωdξ b 2 i' j' ( ω) bij( ω) bx ( ξ ) b ( ξ ) e q' ξp' xqξp ( 2π ) p i 2 ω r2 ξ z 2 r v0 ( xp ). (17) For a fixed ( ω, ξ ) pair, the dreamlet background propagator is a matrix because after interaction with the media, one dreamlet coefficient will spread to other time and space cells. Although the dreamlet propagator is derived from the complex analytic solution of the one way wave equation in the frequency-wavenumber domain, in equation (17), we only take the real part of it. As we mentioned previous, the coefficients of the orthogonal LCB dreamlet is real. This means the wavefield coefficient to be propagated and after propagation is all real. So the propagator matrix needs to be real too, because the imaginary part of the propagator will take no effect at all. In order to visualize the coefficient distribution in the propagator matrix, we rearrange the dreamlet propagator values to see the propagator patterns in the time-space grid as shown in Figure 3. The background velocity is 2000 m/s. All the values concentrate on two bands in the matrix and these locations obey the travel time-distance relation. In Figure 3, each small sub-matrix represents the cross-coupling between different ( ω, ξ) components at that time-space location (, tx, ) as can be seen in the three zoomed-in sub-matrices. Imaging condition in source-receiver Dreamlet survey sinking The Dreamlet operater sinks the whole survey (, s r, ) u t x x to depth z and the image is extracted by evaluating the wavefield amplitude at zero time when the sources coincide with the receivers: s g I ( zx, ) = u( t= 0, x = xx, = x). (18) z 15

16 An intuitive justification of this imaging condition is that the reflection received at the surface was downward continued to the corresponding reflector at zero time and focus best at the zero offset (Claerbout, 1985). In shot-profile migration (Figure 4a), the source wavefield (starting from a virtual wavelet at the surface) and the received reflections are downward continued independently. At very depth and all locations, the image is obtained by the crosscorrelating of the two wavefields. Survey sinking migration downward continues the whole prestack data to each depth level (illustrating in Figure 4b). At each depth, dreamlet source-receiver survey sinking first downward continues the common beamlet source gathers (same as the receiver side in shot-profile dreamlet migration) and then downward continues the propagated wavefield in common beamlet receiver gathers. Therefore, source-receiver migration applies the single square root operator twice, once for each common gathers. Although different in concept and basic principles, shot-profile migration and source-receiver survey sinking migration is equivalent in theory and will produce similar imaging results and common image gathers (Biondi, 2003). In the survey sinking migration, a focused reflection will be generated at zero time and zero offset as the sources and receivers are downward continued to the depth where the reflector located. After focusing, those signals should not propagate anymore because they will defocus and have no contribution to image the structures below. One best feature of the dreamlet propagator is that it gets rid of the used data automatically. We use a three scatters in a homogenous background model to demonstrate this property. Common source seismograms are extracted in different depth for the dreamlet survey sinking using the sunk data, which only keeps the data for the structures below the survey 16

17 system, shown in left column of Figure 6. For comparison, in the same depth, we also put the seismogram from the same source of the shot-profile dreamlet migration (in the middle column) and survey sinking using all the data (in the right column), which imitates what is happening in the frequency domain migration scheme. In Figure 6, the first and the last rows are seismograms at the first and last depth; the second and third rows are the seismograms at the depth where the first and third scatters located. During the survey sinking process, the hyperbolas for each scatter are focused when they reach their depth, whereafter they defocus again. After the focus point, these events do not give contribution to the scatters below. From these figures, we see that the dreamlet propagator always moves the seismogram towards the zero time (at the bottom of the time space panel) and does not wrap around the used signal to the beginning. So the time axis of the sunk data will be shorter and shorter as the depth increased. At every imaging point, the survey sinking will abandon all the used data automatically. This is one aspect in which the dreamlet survey sinking will speed up the migration computation. NUMERICAL TEST ON SEG/EAGE SALT MODEL The SEG/EAGE A-A model is used to demonstrate this method on complex geological structures. The acquisition has 325 shots with left-hand-side receivers and the maximum number of receivers for one shot is 176. The original velocity model has 1200 samples in the horizontal extend and 150 samples in depth, both the sample intervals are 80 feet. The dreamlet survey sinking method is summarized as the following steps:, s r u t x, x to get the wavefield dreamlet (a) Dreamlet decomposition of the wavefiel ( ) coefficients u( t j, ωi, xn, ξ m, xq, ξ p). (b) For each depth, first downward extrapolate the common beamlet source wavefield using dreamlet propagator 17

18 xn ', m ' t j, ωi, xq, ξ p ( j', ωi', q', ξ p' ; ) u t x z+δ z = ξ P 0S SSR ( ) ( t, ω, x, ξ ; t, ω, x, ξ ) u t, ω, x, ξ j' i' q' p' j i q p j i n xn, ξ m m (c) Then sink the common beamlet receivers for the already downward continued wavefield using the dreamlet propagator again xq '', p '' t j', i', xq', p ' ( j'', ωi'', n'', ξ m'' ; ) u t x z+δ z = ω ξ ξ P 0R SSR ( ) ( t, ω, x, ξ ; t, ω, x, ξ ) u t, ω, x, ξ j'' i'' n'' m'' j' i' n' m' j' i' n' xq ', ξ m' p ' (d) Transform the wavefield to the space-frequency domain for the local phase screen correction. (e) Apply the imaging condition at each depth, extract the wavefield amplitude at zero time zero offset as the image amplitude. The operators in step (b) and (c) are same but apply to different gathers with different reference velocities. For computation efficiency, the propagator matrix is precomputed, tabularized with different velocities and used repeatedly during the migration. The local perturbation correction is implemented in the frequency domain (, s r u x, x ) The local perturbation correction terms in equation (13) can be written as.. ω. S1 SSR 1 1 iω ( ) Δz r r v( x ) v0 ( x p ) P = e. (19) The total local perturbation compensation for the survey sinking migration is iω ( + ) Δz s r s r 1 S1 R1 v( x ) v( x ) v0( xn) v0( xp) DSR = SSR SSR = e P P P. (20) We detail the local phase screen correction algorithms as follows: (a) For every frequency multiply the wavefield (, s r u ω x, x ) with the phase-shift using the actual velocity, 1 1 i ( ) z s r v( x ) v( x ) e ω + Δ. s r Where vx ( ) and vx ( ) correspond to the source and receiver location velocities, respectively. (b) Fold the wavefield using the bell function along the common source and common receiver axes (a step of LCB transform). 18

19 (c) Subtract the background phase-shift using the local background velocities, i.e. multiplying 1 1 iω ( + ) Δz s r v0( xn ) v0( xp) e (d) Unfold the wavefield (a step of inverse LCB transform). The images are barely discernible to the eye with the dreamlet survey sinking using the sunk data (Figure 7a) and full data (Figure 7c). Figure 7b is the dreamlet shot-profile imaging result for comparison. Generally speaking, shot-profile and survey sinking migration have similar imaging quality. The 175th shot receiver side seismogram from shot domain dreamlet prestack depth migration are shown in the middle column of Figure 8 (b, e, h, k) and the corresponding common source seismogram from the survey sinking dreamlet using sunk data are in the left column (Figure 8a, d, g, j) and survey sinking using full data are in the right column (Figure 8c, f, i, l). We see that at the deeper depth, the signal does not contribute to the image but still dominates the time space panel. We use the minimum and maximum velocity in the model to evaluate the traveltime and cut the seismogram in shot-profile migration for computation efficiency. So there are truncations in the seismograms. Figure 9 shows the comparison of dreamlet coefficient amounts between the survey sinking and shot-profile migration at every depth for the SEG/EAGE model. The black solid line and the green solid line stand for the survey sinking dreamlet coefficients using the sunk data and the full data, respectively. These two lines indicate the coefficients amount dreamlet survey sinking scheme abandoned during migration. The red dash line and the blue dash line are source and receiver wavefield dreamlet coefficients in the shot domain, and the purple dash line, their sum.. 19

20 We see that at the beginning of the migration, the survey sinking dreamlet coefficients have a larger amount than the shot domain. This is because of the interpolation during the source sinking process in the shallow depth. After downward sinking, the source wavefield will spread and produce more equivalent sources. That means the common source or receiver gathers will increase after sinking. However, after sinking to greater depths, the dreamlet coefficients are drastically decreased in the survey sinking scheme. This is why the survey sinking migration is efficient than the shot domain migration. CONCLUSIONS We combine the idea of dreamlet migration and survey sinking. The dreamlet survey sinking process only keeps the data having physical contributions to the structures beneath the survey system. The dreamlet propagator does not wrap around the seismic data on the time axis and the used seismic data is abandoned automatically during the migration process. The dreamlet source-receiver survey sinking migration is based on the local perturbation theory and can handle the large lateral velocity variation. Tests for prestack depth migration using the synthetic data of the 2D SEG/EAGE model demonstrate the good imaging quality of this method. Moreover, we investigate the computation feature and the dreamlet coefficients change during the depth migration. The wavefield compression in survey sinking is an important issue but not discussed in this paper. This method also can be easily extended to three-dimensional case. ACKNOWLEDGMENTS 20

21 The authors thank Xiaobi Xie, Yueming Ye, Haoran Ren at UCSC and Charles C. Mosher and Jun Cao from ConocoPhillips for helpful discussion on the concept of Survey Sinking. And thank Rui Yan for her help generating the three-scatter model. This work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz. The first author is also partial supported by the China Scholarship Council. REFERENCES Alkhalifah, T. and Fomel S.,2010, Source-receiver two-way wave extrapolation for prestack exploding-reflector modeling and migration: 80th Annual International Meeting, SEG, Expanded Abstracts, Auscher, P., 1994, Remarks on the local Fourier bases, in J.J. Benedetto and M. W. Frazier, eds., Wavelets, mathematics, and applications: CRC Press, Balian, R., 1981, Un principle d incertitude en theorie du signal ou en mecanique quantique, Comptes Rendus de l Academie des Sciences, Paris, Serie II, 292, Biondi, B., 2003, Equivalence of source-receiver migration and shot-profile migration: Geophysics, 68, Biondi, B., 2006, 3D seismic imaging: SEG. Candès, E., and D. L. Donoho, 2002, New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities: Communications on Pure and Applied Mathematice, 57,

22 Candès, E., and L. Demanet, 2005, The curvelet representation of wave propagators is optimally sparse: Communications on Pure and Applied Mathematice, 58, Candès, E.J. and L. Demanet, 2005, The curvelet representation of wave propagators is optimally sparse: Comm. Pure Appl. Math , Chauris, H., and T. Nguyen, 2008, Seismic demigration/migration in the curvelet domain: Geophysics, 73, S35-S46. Chen, L. and Wu, R.S., 2002, Target-oriented prestack beamlet migration using Gabor- Daubechies frames, Expanded abstracts, SEG 72nd Annual Meeting, Chen, L. and Wu, R.S., 2006, Target-oriented beamlet migration based on Gabor- Daubechies frame decomposition: Geophysics, 71, Claerbout, J.F., 1985, Imaging the earth s interior: Blackwell Scientific Publications. Coifman,R.R. and Meyer, Y., 1991, Remarques sur l analyse de Fourier a fenetre, Comptes Rendus de l Academie des Sciences, Paris, Serie I 312, Daubechies, I., 1990, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Info. Theory, 36, Daubechies, I., 1992, Ten Lectures on Wavelets, Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics. de Hoop, M., Rousseau, J., and Wu, R.S., 2000, General formulation of the phase-screen approximation for the scattering of acoustic waves, Wave Motion,31, Douma, H., and M. V. de Hoop, 2007, Leading-order seismic imaging using curvelets: Geophysics, 72, S231-S248. Fishman, L. and McCoy, J.J., 1984, Derivation and application of extended parabolic wave theories II. Path integral representations, J. Math. Phys., 25,

23 Jin, S., and Wu, R.S., 1999, Depth migration with a windowed screen propagator, Journal of Seismic Exploration, 8(1), Jin, S., Mosher, C.C. and Wu, R.S., 2002, Offset-domain pseudo-screen prestack depth migration: Geophysics, 67, Luo, M. and Jin, S., 2006, Offset-domain LCB beamlet prestack depth migration: 76th Annual International Meeting, SEG, Expanded Abstracts, Luo, M. and Wu, R.S., 2003, 3D beamlet prestack depth migration using the local cosine basis propagator, Expanded abstracts, SEG 73rd Annual Meeting, Ma, Y., and G. F. Margrave, 2008, Seismic depth imaging with the Gabor transform: Geophysics, 73, Mallat, S., 1999, A Wavelet Tour of Signal Processing, second edition, Academic Press. Malvar, H.S., 1992, Signal processing with lapped transforms, Artech House, Norwood, MA. Mao, J., and R. S. Wu, 2007, Illumination analysis using local exponential beamlets: 77th Annual International Meeting, SEG, Expanded Abstracts, Pedersen, Ø., Brandsberg-Dahl, S., and Ursin, B., 2010, Seismic imaging using lateral adaptive windows: Geophysics, 75, Ristow, D., and Ruhl, T., 1994, Fourier finite-difference migration: Geopysics, 59, Sandberg, K. and Beylkin G., 2009, Full-wave-equation depth extrapolation for migration: Geophysics, 74,

24 Steinberg, B.Z., 1993, Evolution of local spectra in smoothly varying nonhomogeneous environments-local canonization and marching algorithms, J. Acoust. Soc. Am. 93, Stoffa, P. L., Fokkema, J. T., de Luna Freire, R. M., and Kessinger, W. P., 1990, Splitstep Fourier migration: Geophysics, 55, Wang, Y. and Wu, R.S., 1998, Migration operator decomposition and compression using a new wavelet packet best basis algorithm, Expanded abstracts, SEG 68th Annual Meeting, Wang, Y. and Wu, R.S., 2002, Beamlet prestack depth migration using local cosine basis propagator, Expanded abstracts, SEG 72nd Annual Meeting, Wang, Y., Cook, R., and Wu, R.S., 2003, 3D local cosine beamlet propagator, Expanded abstracts, SEG 73th Annual Meeting, Wang, Y., Verm, R., and Bednar, B., 2005, Application of beamlet migration to the SmaartJV Sigsbee2A model, Expanded abstracts, SEG 75th Annual Meeting, Wickerhauser, M.V., 1994, Adapted wavelet analysis form theory to software: A K Peters. Wu, B.Y. and Wu, R.S., 2010, Orthogonal dreamlet decomposition and its application to seismic imaging: SEG 80th Annual International Meeting, Expanded Abstracts, Wu, R. S. and Chen, L., 2002b, Mapping directional illumination and acquisitionaperture efficacy by beamlet propagators: 72nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts,

25 Wu, R. S., and Chen, L, 2001, Beamlet migration using Gabor-Daubechies frame propagator: 63rd Conference & Technical Exhibition, EAGE, Expanded abstracts, P.74. Wu, R. S., and Chen, L., 2002a, Wave propagation and imaging using Gabor-Daubechies beamlets: Theoretical and Computational Acoustics, World Scientific, New Jersey, Wu, R. S., and J. Mao, 2007, Beamlet migration using local harmonic bases: 77th Annual International Meeting, SEG, Expanded Abstracts, Wu, R. S., B. Y. Wu, and Y. Geng, 2008, Seismic wave propagation and imaging using time-space wavelets, 78th Annual International Meeting, SEG, Expanded Abstracts, Wu, R. S., Chen, L., and Wang, Y., 2002, Prestack migration/imaging using synthetic beamsources and plane sources, Stud. Geophys. Geod., 46, 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, Wu, R.S. and F. Yang, 1997, Seismic imaging in wavelet domain: Decomposition and compression of imaging operator, "Wavelet Applications in Signal and Image Processing V", Proc. SPIE, 3169, Wu, R.S. and Wang, Y., 1998, Comparison of propagator decomposition in seismic imaging by wavelets, wavelet-packets, and local harmonics, Mathematical Methods in Geophysical Imaging, V, Proc. SPIE, 3453,

26 Wu, R.S. and Xie, X.B., 1994, Multi-screen backpropagator for fast 3D elastic prestack migration: Mathematical Methods in Geophysical Imaging II, SPIE vol. 2301, Xie, X.B. and Wu, R.S., 1998, Improve the wide angle accuracy of screen method under large contrast, Expanded abstracts, SEG 68th Annual Meeting, Wu, R.S., B. Wu, and Y. Geng, 2009, Imaging in compressed domain using dreamlets, CPS/SEG Beijing '2009, International Geophysical Conference, Expanded Abstracts, ID: 57. Wu, R.S., F. Yang, Z. Wang and L. Zhang, 1997, Migration Operator Compression by Wavelet Transform: Beamlet Migrator, Expanded Abstracts of the Technical Program, SEG 67th Annual Meeting, Wu, R.S., Wang, Y. and Luo, M., 2003, Local-cosine beamlet migration for 3D complex structures, Eighth International Congress of the Brazilian Geophysical Society, Wu, R.S., Wang, Y. and Luo, M., 2008, Beamlet migration using local cosine basis: Geophysics, 73,

27 Figure 1 Schematic illustration of the dreamlet survey sinking wavefield resorting for the SEG/EAGE salt model acquisition system. The time axis is perpendicular to the sourcereceiver plane. And the blue part is zero padding for the calculation aperture. The strip in the middle of the plane is the original seismic data distribution. 27

28 Figure 2. Source-receiver Dreamlet coefficients for the 2D SEG/EAGE model. (a, b, c) are for the common beamlet sources and (d, e, f) for the common beamlet receivers. Coefficients are chosen at low wavenumbers (a, d), at mid range (b, e), and large wavenumbers (c, f). 28

29 Figure 2. Source-receiver Dreamlet coefficients for the 2D SEG/EAGE model. (a, b, c) are for the common beamlet sources and (d, e, f) for the common beamlet receivers. Coefficients are chosen at low wavenumbers (a, d), at mid range (b, e), and large wavenumbers (c, f). 29

30 Figure 3: Coefficient distribution in the propagator matrix of the Dreamlet propagator. In the figure, each small sub-matrix represents the cross-coupling between different ( ω, ξ) components at that time-space location (, tx, ) as can be seen in the three zoomed-in submatrices. 30

31 Figure 4. Schematic illustration of the shot-profile migration (a) and the source-receiver survey sinking migration (b). In shot-profile migration, the source and receiver wavefield downward continues independently and the source-receiver survey sinking migrates the whole data to each depth. 31

32 Figure 5. A three-scatter model used to demonstrate the property of the dreamlet propagator. 32

33 Figure 6. The common source seismogram from survey sinking using sunk data (left column), from shot domain (middle column) and from survey sinking using all data (right column). The first row (a, b, c) and last row (j, k, l) are from the first and the last depth. The second row (d, e, f) and third row (g, h, i) are from the depths where the first and third scatters located. 33

34 Figure 6. The common source seismogram from survey sinking using sunk data (left column), from shot domain (middle column) and from survey sinking using all data (right column). The first row (a, b, c) and last row (j, k, l) are from the first and the last depth. The second row (d, e, f) and third row (g, h, i) are from the depths where the first and third scatters located. 34

35 Figure 6. The common source seismogram from survey sinking using sunk data (left column), from shot domain (middle column) and from survey sinking using all data (right column). The first row (a, b, c) and last row (j, k, l) are from the first and the last depth. The second row (d, e, f) and third row (g, h, i) are from the depths where the first and third scatters located. 35

36 Figure 6. The common source seismogram from survey sinking using sunk data (left column), from shot domain (middle column) and from survey sinking using all data (right column). The first row (a, b, c) and last row (j, k, l) are from the first and the last depth. The second row (d, e, f) and third row (g, h, i) are from the depths where the first and third scatters located. 36

37 Figure 7 (a) Dreamlet survey sinking migration result using the sunk data; (b) is the image by shot domain dreamlet method; (c) dreamlet survey sinking using full data. 37

38 Figure 8. In left column are the common source seismograms from survey sinking migration using sunk data (a, d, g, j) and in the right column are seismograms for using full data (c, f, i, l). The middle column is the same source from shot domain migration (b, e, h, k). The first and the last rows are from first and last depth. The second and third rows are from depth 50 and

39 Figure 8. In left column are the common source seismograms from survey sinking migration using sunk data (a, d, g, j) and in the right column are seismograms for using full data (c, f, i, l). The middle column is the same source from shot domain migration (b, e, h, k). The first and the last rows are from first and last depth. The second and third rows are from depth 50 and

40 Figure 8. In left column are the common source seismograms from survey sinking migration using sunk data (a, d, g, j) and in the right column are seismograms for using full data (c, f, i, l). The middle column is the same source from shot domain migration (b, e, h, k). The first and the last rows are from first and last depth. The second and third rows are from depth 50 and

41 Figure 8. In left column are the common source seismograms from survey sinking migration using sunk data (a, d, g, j) and in the right column are seismograms for using full data (c, f, i, l). The middle column is the same source from shot domain migration (b, e, h, k). The first and the last rows are from first and last depth. The second and third rows are from depth 50 and

42 Figure 9. Variation of dreamlet coefficient amount during migration. The black solid line and green solid line are for the survey sinking dreamlet coefficients using sunk data and the full data, respectively; the red dash line and the blue dash line are the source and receiver dreamlet coefficients in the shot domain, and the purple dash line, the sum in the shot domain. 42